Procedurally Generating a Non-Contiguous Mesh

I'm generating a mesh as a "grid" of individual cells. Each of these cells can move independently. I'm able to generate the mesh and update the vertices to move the individual cells, but it leaves a stretched-out texture between adjacent cells when they're not next to each other.

Is there a way to procedurally generate a mesh and move pieces independently? (while keeping the performance benefits of generating a mesh over using a separate MeshRenderer for each cell)

It's also completely possible I've done something terribly wrong in my code. I've tried generating the entire grid as a single mesh, and the current version creates a mesh for each cell and combines them using CombineMeshes. Both have had about the same results. This is the code I'm using to generate the mesh for an individual cell:

 var meshVertices = new List<Vector3>();
 var meshNormals = new List<Vector3>();
 var meshUVs = new List<Vector2>();
 var meshTris = new List<int>();

 //Cell vertices
 meshVertices.Add(new Vector3(xMin, 0, zMin));
 meshVertices.Add(new Vector3(xMax, 0, zMin));
 meshVertices.Add(new Vector3(xMin, 0, zMax));
 meshVertices.Add(new Vector3(xMax, 0, zMax));

 //Normals for each vertex
 meshNormals.Add(Vector3.up);
 meshNormals.Add(Vector3.up);
 meshNormals.Add(Vector3.up);
 meshNormals.Add(Vector3.up);

 //UVs for each vertex
 meshUVs.Add(new Vector2(0, 0));
 meshUVs.Add(new Vector2(1, 0));
 meshUVs.Add(new Vector2(0, 1));
 meshUVs.Add(new Vector2(1, 1));

 //Vertex indices for the triangles
 meshTris.Add(0);
 meshTris.Add(2);
 meshTris.Add(3);
 meshTris.Add(0);
 meshTris.Add(3);
 meshTris.Add(1);

 var mesh = new Mesh
         {
             vertices = meshVertices.ToArray(),
             triangles = meshTris.ToArray(),
             normals = meshNormals.ToArray(),
             uv = meshUVs.ToArray(),
         };

 return new CombineInstance
            {
                mesh = mesh,
            };

Once all the cells are created, I combine them like so:

 var mesh = new Mesh();
 mesh.CombineMeshes(cellCombineInstances.ToArray());
 mesh.RecalculateBounds();

 var meshObject = new GameObject("GridMesh");
 meshObject.transform.parent = this.transform;
 meshObject.transform.position = this.transform.position;
 meshObject.transform.localScale = Vector3.one;
 meshObject.transform.localRotation = Quaternion.identity;
 meshObject.transform.localPosition = Vector3.zero;

 var mat = new Material(this.CellMaterial);
 meshObject.AddComponent<MeshRenderer>();
 meshObject.renderer.material = mat;

 var meshFilter = meshObject.AddComponent<MeshFilter>();
 meshFilter.mesh = mesh;

When I move the vertices, I use:

 for (var i = 0; i < 4; i++)
 {
     //startVertex is determined elsewhere, but it's the index of the first vertex for an individual cell
     var vertex = vertices[startVertex + i];
     //The x and z coordinates can also change, but for now I'm only concerned with the y-axis.
     vertex.y = newPosition.y;
     vertices[startVertex + i] = vertex;
 }