Numbers - Move Cell Contents WITHOUT altering background color

There is not way I am aware of when using that method to move cells.

If the cells to be moved are not being referenced in any formulas and do not contain any formulas, you can do this:

1. Select the range of cells
2. Copy
3. Select the top left cell of the destination (or all of the destination cells)
4. Paste and Match Style
5. Go back and select the originating cells again
6. Delete (backspace)

This will preserve the formatting of the source cells and the destination cells.

If you want to preserve the formatting of the source cells and have the same formatting in the destination cells and the cells are not referenced in formulas and do not contain formulas,

1. Lift, drag and drop the cells like usual but hold Option while doing so
2. Select the original cells again
3. Delete

This will bring the contents and formatting to the destination while preserving the formatting of the original cells.

If the cells are referenced in formulas or if they contain formulas, you will have to do the drag and drop method to move them or the formulas will/may get messed up. The best I have been able to do is preserve the formatting of either the source or the destination but not both.

1. Select the source cells (or destination cells, whichever you want to keep the formatting for).
2. Copy Style
3. Lift, drag, and drop the cells to the destination like usual
4. Select the cells from step 1
5. Paste Style

One more idea.

1. Select the range of cells for which you want to preserve the formatting. If your source cells are adjacent to the destination cells you should be able to select just those groups of cells as a range but, if not, include the cells between them so that you have a range selected.
2. Copy Style
3. Lift, drag, and drop the cells like you normally would to their new location.
4. Select all the same cells from step 1
5. Paste Style

I've not tested this or the other ideas under a lot of conditions, just with random cell fill colors. Hopefully they work with other borders and other formatting settings too.