Hope you figured it out by now but here goes.
Two methods I have used in past. Both need layer counts to match. electrical and documentation.
Ascii
Asci out board to copy without board outline then import into receiving board. refdes name and positions will remain same as copied board. Net names will be preserved.
Copy and paste
Open board to copy. Select parts, traces, vias. Move curser to 0,0 then copy. Close board Open board to paste into. Activate ECO mode. Paste will occur at cursor location. All pasted refdes will increment up from existing. Be located relative to curser position. All pasted nets will be added as $$$# generics.
Could also be done with reuse, but I have not used that enough to know what you will run into or how to set up.
William