edit menu, copy merged. paste and move layer at will.
copy, not cut. You'd have to go through each layer deleting the selection before pasting if you wanted to leave a hole.
Why is it like that? well, what you "move" is actually a temporary layer that your selection has been cut to. This temporary layer is placed on top and will be merged with the layer you moved from when you're done.
When dealing with multiple layers, this is ambiguous. So it's just not possible.