What do you mean by ‘copper layers’?
Legacy canvas mode (F9) has a block selection operation where you can choose what goes into that group.
If you then also (after the selection dialog closes) don’t move your mouse and use the relative coordinate readout, reset the readout with [Space] to zero and use the keyboard arrow keys for moving the selection you can achieve what you want.
The grid would then need to be set to the smallest denominator you want to be able to move.
And after you got it moved where it should be, just left click with the mouse without moving it.
You got a screenshot that would be suited to explain your aim a bit more clearly?
Another option is to create (at least for the mechanical side of things) a footprint that contains the holes, outline, aso.. and use that as reference that can be moved without changing relative mechanical coordinates.
See this thread: