diff --git a/src/array-utils.ts b/src/array-utils.ts
new file mode 100644
index 0000000..cf37e4c
--- /dev/null
+++ b/src/array-utils.ts
@@ -0,0 +1,38 @@
+/*
+Copyright 2023 New Vector Ltd
+
+Licensed under the Apache License, Version 2.0 (the "License");
+you may not use this file except in compliance with the License.
+You may obtain a copy of the License at
+
+    http://www.apache.org/licenses/LICENSE-2.0
+
+Unless required by applicable law or agreed to in writing, software
+distributed under the License is distributed on an "AS IS" BASIS,
+WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+See the License for the specific language governing permissions and
+limitations under the License.
+*/
+
+/**
+ * Gets the index of the last element in the array to satsify the given
+ * predicate.
+ */
+// TODO: remove this once TypeScript recognizes the existence of
+// Array.prototype.findLastIndex
+export function findLastIndex<T>(
+  array: T[],
+  predicate: (item: T) => boolean
+): number | null {
+  for (let i = array.length - 1; i >= 0; i--) {
+    if (predicate(array[i])) return i;
+  }
+
+  return null;
+}
+
+/**
+ * Counts the number of elements in an array that satsify the given predicate.
+ */
+export const count = <T>(array: T[], predicate: (item: T) => boolean): number =>
+  array.reduce((acc, item) => (predicate(item) ? acc + 1 : acc), 0);
diff --git a/src/video-grid/NewVideoGrid.module.css b/src/video-grid/NewVideoGrid.module.css
index 75ee707..7e34a2d 100644
--- a/src/video-grid/NewVideoGrid.module.css
+++ b/src/video-grid/NewVideoGrid.module.css
@@ -18,7 +18,7 @@ limitations under the License.
   contain: strict;
   position: relative;
   flex-grow: 1;
-  padding: 0 20px;
+  padding: 0 20px var(--footerHeight);
   overflow-y: auto;
   overflow-x: hidden;
 }
@@ -28,7 +28,6 @@ limitations under the License.
   display: grid;
   grid-auto-rows: 163px;
   gap: 8px;
-  padding-bottom: var(--footerHeight);
 }
 
 .slot {
@@ -37,7 +36,7 @@ limitations under the License.
 
 @media (min-width: 800px) {
   .grid {
-    padding: 0 22px;
+    padding: 0 22px var(--footerHeight);
   }
 
   .slotGrid {
diff --git a/src/video-grid/NewVideoGrid.tsx b/src/video-grid/NewVideoGrid.tsx
index 0511840..00c0650 100644
--- a/src/video-grid/NewVideoGrid.tsx
+++ b/src/video-grid/NewVideoGrid.tsx
@@ -47,6 +47,8 @@ import {
   forEachCellInArea,
   cycleTileSize,
   appendItems,
+  tryMoveTile,
+  resize,
 } from "./model";
 import { TileWrapper } from "./TileWrapper";
 
@@ -84,8 +86,11 @@ const useGridState = (
         }),
       };
 
-      // Step 2: Backfill gaps left behind by removed tiles
-      const grid2 = fillGaps(grid1);
+      // Step 2: Resize the grid if necessary and backfill gaps left behind by
+      // removed tiles
+      // Resizing already takes care of backfilling gaps
+      const grid2 =
+        columns !== grid1.columns ? resize(grid1, columns!) : fillGaps(grid1);
 
       // Step 3: Add new tiles to the end of the grid
       const existingItemIds = new Set(
@@ -207,14 +212,10 @@ export function NewVideoGrid<T>({
     // eslint-disable-next-line react-hooks/exhaustive-deps
   }, [slotGrid, slotGridGeneration, gridBounds]);
 
-  const [columns] = useReactiveState<number | null>(
-    // Since grid resizing isn't implemented yet, pick a column count on mount
-    // and stick to it
-    (prevColumns) =>
-      prevColumns !== undefined && prevColumns !== null
-        ? prevColumns
-        : // The grid bounds might not be known yet
-        gridBounds.width === 0
+  const columns = useMemo(
+    () =>
+      // The grid bounds might not be known yet
+      gridBounds.width === 0
         ? null
         : Math.max(2, Math.floor(gridBounds.width * 0.0045)),
     [gridBounds]
@@ -283,6 +284,8 @@ export function NewVideoGrid<T>({
   const animateDraggedTile = (endOfGesture: boolean) => {
     const { tileId, tileX, tileY, cursorX, cursorY } = dragState.current!;
     const tile = tiles.find((t) => t.item.id === tileId)!;
+    const originIndex = grid!.cells.findIndex((c) => c?.item.id === tileId);
+    const originCell = grid!.cells[originIndex]!;
 
     springRef.current
       .find((c) => (c.item as Tile).item.id === tileId)
@@ -313,23 +316,36 @@ export function NewVideoGrid<T>({
             }
       );
 
-    const overTile = tiles.find(
-      (t) =>
-        cursorX >= t.x &&
-        cursorX < t.x + t.width &&
-        cursorY >= t.y &&
-        cursorY < t.y + t.height
+    const columns = grid!.columns;
+    const rows = row(grid!.cells.length - 1, grid!) + 1;
+
+    const cursorColumn = Math.floor(
+      (cursorX / slotGrid!.clientWidth) * columns
     );
-    if (overTile !== undefined && overTile.item.id !== tileId) {
-      setGrid((g) => ({
-        ...g!,
-        cells: g!.cells.map((c) => {
-          if (c?.item === overTile.item) return { ...c, item: tile.item };
-          if (c?.item === tile.item) return { ...c, item: overTile.item };
-          return c;
-        }),
-      }));
-    }
+    const cursorRow = Math.floor((cursorY / slotGrid!.clientHeight) * rows);
+
+    const cursorColumnOnTile = Math.floor(
+      ((cursorX - tileX) / tile.width) * originCell.columns
+    );
+    const cursorRowOnTile = Math.floor(
+      ((cursorY - tileY) / tile.height) * originCell.rows
+    );
+
+    const dest =
+      Math.max(
+        0,
+        Math.min(
+          columns - originCell.columns,
+          cursorColumn - cursorColumnOnTile
+        )
+      ) +
+      grid!.columns *
+        Math.max(
+          0,
+          Math.min(rows - originCell.rows, cursorRow - cursorRowOnTile)
+        );
+
+    if (dest !== originIndex) setGrid((g) => tryMoveTile(g, originIndex, dest));
   };
 
   // Callback for useDrag. We could call useDrag here, but the default
diff --git a/src/video-grid/model.ts b/src/video-grid/model.ts
index d913c9b..88aa38e 100644
--- a/src/video-grid/model.ts
+++ b/src/video-grid/model.ts
@@ -17,6 +17,7 @@ limitations under the License.
 import TinyQueue from "tinyqueue";
 
 import { TileDescriptor } from "./VideoGrid";
+import { count, findLastIndex } from "../array-utils";
 
 /**
  * A 1×1 cell in a grid which belongs to a tile.
@@ -105,17 +106,6 @@ export function getPaths(dest: number, g: Grid): (number | null)[] {
   return edges as (number | null)[];
 }
 
-function findLastIndex<T>(
-  array: T[],
-  predicate: (item: T) => boolean
-): number | null {
-  for (let i = array.length - 1; i >= 0; i--) {
-    if (predicate(array[i])) return i;
-  }
-
-  return null;
-}
-
 const findLast1By1Index = (g: Grid): number | null =>
   findLastIndex(g.cells, (c) => c?.rows === 1 && c?.columns === 1);
 
@@ -185,6 +175,8 @@ const areaEnd = (
   g: Grid
 ): number => start + columns - 1 + g.columns * (rows - 1);
 
+const cloneGrid = (g: Grid): Grid => ({ ...g, cells: [...g.cells] });
+
 /**
  * Gets the index of the next gap in the grid that should be backfilled by 1×1
  * tiles.
@@ -209,11 +201,150 @@ function getNextGap(g: Grid): number | null {
   return null;
 }
 
+/**
+ * Gets the index of the origin of the tile to which the given cell belongs.
+ */
+function getOrigin(g: Grid, index: number): number {
+  const initialColumn = column(index, g);
+
+  for (
+    let i = index;
+    i >= 0;
+    i = column(i, g) === 0 ? i - g.columns + initialColumn : i - 1
+  ) {
+    const cell = g.cells[i];
+    if (
+      cell !== undefined &&
+      cell.origin &&
+      inArea(index, i, areaEnd(i, cell.columns, cell.rows, g), g)
+    )
+      return i;
+  }
+
+  throw new Error("Tile is broken");
+}
+
+/**
+ * Moves the tile at index "from" over to index "to", displacing other tiles
+ * along the way.
+ * Precondition: the destination area must consist of only 1×1 tiles.
+ */
+function moveTile(g: Grid, from: number, to: number) {
+  const tile = g.cells[from]!;
+  const fromEnd = areaEnd(from, tile.columns, tile.rows, g);
+  const toEnd = areaEnd(to, tile.columns, tile.rows, g);
+
+  const displacedTiles: Cell[] = [];
+  forEachCellInArea(to, toEnd, g, (c, i) => {
+    if (c !== undefined && !inArea(i, from, fromEnd, g)) displacedTiles.push(c);
+  });
+
+  const movingCells: Cell[] = [];
+  forEachCellInArea(from, fromEnd, g, (c, i) => {
+    movingCells.push(c!);
+    g.cells[i] = undefined;
+  });
+
+  forEachCellInArea(
+    to,
+    toEnd,
+    g,
+    (_c, i) => (g.cells[i] = movingCells.shift())
+  );
+  forEachCellInArea(
+    from,
+    fromEnd,
+    g,
+    (_c, i) => (g.cells[i] ??= displacedTiles.shift())
+  );
+}
+
+/**
+ * Moves the tile at index "from" over to index "to", if there is space.
+ */
+export function tryMoveTile(g: Grid, from: number, to: number): Grid {
+  const tile = g.cells[from]!;
+
+  if (
+    to > 0 &&
+    to < g.cells.length &&
+    column(to, g) <= g.columns - tile.columns
+  ) {
+    const fromEnd = areaEnd(from, tile.columns, tile.rows, g);
+    const toEnd = areaEnd(to, tile.columns, tile.rows, g);
+
+    // The contents of a given cell are 'displaceable' if it's empty, holds a
+    // 1×1 tile, or is part of the original tile we're trying to reposition
+    const displaceable = (c: Cell | undefined, i: number): boolean =>
+      c === undefined ||
+      (c.columns === 1 && c.rows === 1) ||
+      inArea(i, from, fromEnd, g);
+
+    if (allCellsInArea(to, toEnd, g, displaceable)) {
+      // The target space is free; move
+      const gClone = cloneGrid(g);
+      moveTile(gClone, from, to);
+      return gClone;
+    }
+  }
+
+  // The target space isn't free; don't move
+  return g;
+}
+
+/**
+ * Attempts to push a tile upwards by one row, displacing 1×1 tiles and shifting
+ * enlarged tiles around when necessary.
+ * @returns Whether the tile was actually pushed
+ */
+function pushTileUp(g: Grid, from: number): boolean {
+  const tile = g.cells[from]!;
+
+  // TODO: pushing large tiles sideways might be more successful in some
+  // situations
+  const cellsAboveAreDisplacable =
+    from - g.columns >= 0 &&
+    allCellsInArea(
+      from - g.columns,
+      from - g.columns + tile.columns - 1,
+      g,
+      (c, i) =>
+        c === undefined ||
+        (c.columns === 1 && c.rows === 1) ||
+        pushTileUp(g, getOrigin(g, i))
+    );
+
+  if (cellsAboveAreDisplacable) {
+    moveTile(g, from, from - g.columns);
+    return true;
+  } else {
+    return false;
+  }
+}
+
 /**
  * Backfill any gaps in the grid.
  */
 export function fillGaps(g: Grid): Grid {
-  const result: Grid = { ...g, cells: [...g.cells] };
+  const result = cloneGrid(g);
+
+  // This will hopefully be the size of the grid after we're done here, assuming
+  // that we can pack the large tiles tightly enough
+  const idealLength = count(result.cells, (c) => c !== undefined);
+
+  // Step 1: Take any large tiles hanging off the bottom of the grid, and push
+  // them upwards
+  for (let i = result.cells.length - 1; i >= idealLength; i--) {
+    const cell = result.cells[i];
+    if (cell !== undefined && (cell.columns > 1 || cell.rows > 1)) {
+      const originIndex =
+        i - (cell.columns - 1) - result.columns * (cell.rows - 1);
+      // If it's not possible to pack the large tiles any tighter, give up
+      if (!pushTileUp(result, originIndex)) break;
+    }
+  }
+
+  // Step 2: Fill all 1×1 gaps
   let gap = getNextGap(result);
 
   if (gap !== null) {
@@ -263,9 +394,6 @@ export function fillGaps(g: Grid): Grid {
     } while (gap !== null);
   }
 
-  // TODO: If there are any large tiles on the last row, shuffle them back
-  // upwards into a full row
-
   // Shrink the array to remove trailing gaps
   const finalLength =
     (findLastIndex(result.cells, (c) => c !== undefined) ?? -1) + 1;
@@ -290,6 +418,11 @@ export function appendItems(items: TileDescriptor<unknown>[], g: Grid): Grid {
   };
 }
 
+const largeTileDimensions = (g: Grid): [number, number] => [
+  Math.min(3, Math.max(2, g.columns - 1)),
+  2,
+];
+
 /**
  * Changes the size of a tile, rearranging the grid to make space.
  * @param tileId The ID of the tile to modify.
@@ -305,9 +438,7 @@ export function cycleTileSize(tileId: string, g: Grid): Grid {
 
   // The target dimensions, which toggle between 1×1 and larger than 1×1
   const [toWidth, toHeight] =
-    fromWidth === 1 && fromHeight === 1
-      ? [Math.min(3, Math.max(2, g.columns - 1)), 2]
-      : [1, 1];
+    fromWidth === 1 && fromHeight === 1 ? largeTileDimensions(g) : [1, 1];
 
   // If we're expanding the tile, we want to create enough new rows at the
   // tile's target position such that every new unit of grid area created during
@@ -381,13 +512,18 @@ export function cycleTileSize(tileId: string, g: Grid): Grid {
 
   // Copy tiles from the original grid to the new one, with the new rows
   // inserted at the target location
-  g.cells.forEach((c, src) => {
+  g.cells.forEach((c, from) => {
     if (c?.origin && c.item.id !== tileId) {
       const offset =
-        row(src, g) > toRow + candidateHeight - 1 ? g.columns * newRows : 0;
-      forEachCellInArea(src, areaEnd(src, c.columns, c.rows, g), g, (c, i) => {
-        gappyGrid.cells[i + offset] = c;
-      });
+        row(from, g) > toRow + candidateHeight - 1 ? g.columns * newRows : 0;
+      forEachCellInArea(
+        from,
+        areaEnd(from, c.columns, c.rows, g),
+        g,
+        (c, i) => {
+          gappyGrid.cells[i + offset] = c;
+        }
+      );
     }
   });
 
@@ -414,3 +550,46 @@ export function cycleTileSize(tileId: string, g: Grid): Grid {
   // Fill any gaps that remain
   return fillGaps(gappyGrid);
 }
+
+/**
+ * Resizes the grid to a new column width.
+ */
+export function resize(g: Grid, columns: number): Grid {
+  const result: Grid = { columns, cells: [] };
+  const [largeColumns, largeRows] = largeTileDimensions(result);
+
+  // Copy each tile from the old grid to the resized one in the same order
+
+  // The next index in the result grid to copy a tile to
+  let next = 0;
+
+  for (const cell of g.cells) {
+    if (cell?.origin) {
+      const [nextColumns, nextRows] =
+        cell.columns > 1 || cell.rows > 1 ? [largeColumns, largeRows] : [1, 1];
+
+      // If there isn't enough space left on this row, jump to the next row
+      if (columns - column(next, result) < nextColumns)
+        next = columns * (Math.floor(next / columns) + 1);
+      const nextEnd = areaEnd(next, nextColumns, nextRows, result);
+
+      // Expand the cells array as necessary
+      if (result.cells.length <= nextEnd)
+        result.cells.push(...new Array(nextEnd + 1 - result.cells.length));
+
+      // Copy the tile into place
+      forEachCellInArea(next, nextEnd, result, (_c, i) => {
+        result.cells[i] = {
+          item: cell.item,
+          origin: i === next,
+          columns: nextColumns,
+          rows: nextRows,
+        };
+      });
+
+      next = nextEnd + 1;
+    }
+  }
+
+  return fillGaps(result);
+}
diff --git a/test/video-grid/model-test.ts b/test/video-grid/model-test.ts
index 7093ca7..fe0364c 100644
--- a/test/video-grid/model-test.ts
+++ b/test/video-grid/model-test.ts
@@ -21,7 +21,9 @@ import {
   fillGaps,
   forEachCellInArea,
   Grid,
+  resize,
   row,
+  tryMoveTile,
 } from "../../src/video-grid/model";
 import { TileDescriptor } from "../../src/video-grid/VideoGrid";
 
@@ -169,6 +171,50 @@ dddd
 iegh`
 );
 
+testFillGaps(
+  "keeps a large tile from hanging off the bottom",
+  `
+abcd
+efgh
+    
+ii  
+ii`,
+  `
+abcd
+iigh
+iief`
+);
+
+testFillGaps(
+  "pushes a chain of large tiles upwards",
+  `
+abcd
+e fg
+hh  
+hh  
+ ii 
+ ii`,
+  `
+hhcd
+hhfg
+aiib
+eii`
+);
+
+testFillGaps(
+  "gives up on pushing large tiles upwards when not possible",
+  `
+aabb
+aabb
+cc  
+cc`,
+  `
+aabb
+aabb
+cc  
+cc`
+);
+
 function testCycleTileSize(
   title: string,
   tileId: string,
@@ -281,3 +327,112 @@ def`;
   );
   expect(showGrid(appendItems(newItems, mkGrid(grid1)))).toBe(grid2);
 });
+
+function testTryMoveTile(
+  title: string,
+  from: number,
+  to: number,
+  input: string,
+  output: string
+): void {
+  test(`tryMoveTile ${title}`, () => {
+    expect(showGrid(tryMoveTile(mkGrid(input), from, to))).toBe(output);
+  });
+}
+
+testTryMoveTile(
+  "refuses to move a tile too far to the left",
+  1,
+  -1,
+  `
+abc`,
+  `
+abc`
+);
+
+testTryMoveTile(
+  "refuses to move a tile too far to the right",
+  1,
+  3,
+  `
+abc`,
+  `
+abc`
+);
+
+testTryMoveTile(
+  "moves a large tile to an unoccupied space",
+  3,
+  1,
+  `
+a b
+ccd
+cce`,
+  `
+acc
+bcc
+d e`
+);
+
+testTryMoveTile(
+  "refuses to move a large tile to an occupied space",
+  3,
+  1,
+  `
+abb
+ccd
+cce`,
+  `
+abb
+ccd
+cce`
+);
+
+function testResize(
+  title: string,
+  columns: number,
+  input: string,
+  output: string
+): void {
+  test(`resize ${title}`, () => {
+    expect(showGrid(resize(mkGrid(input), columns))).toBe(output);
+  });
+}
+
+testResize(
+  "contracts the grid",
+  2,
+  `
+abbb
+cbbb
+ddde
+dddf
+gh`,
+  `
+af
+bb
+bb
+ch
+dd
+dd
+eg`
+);
+
+testResize(
+  "expands the grid",
+  4,
+  `
+af
+bb
+bb
+ch
+dd
+dd
+eg`,
+  `
+bbbc
+bbbf
+addd
+hddd
+ge`
+);