AFAIK there’s no way to move both the schematic and board compontents (with coordinate/rotation information).
What I typically do is just create copies of the .brd and .sch files in FileExplorer, rename them to match my new version, and start modifying those. But this doesn’t really help you create a new design from a “library” of schematic/layout pieces.
This may be something that is possible with other, probably more expensive, CAD packages.
I can copy and paste PCB designs, and parts of them, from one PCB to another without any problems using Pulsonix. It’ not usually a good idea because the schematic won’t match the PCB, and ‘PCB Safe Mode’ has to be disabled.
I can also ‘apply’ a layout pattern from one PCB to another one using the same components. This is a much better way of doing it.