Cut, Copy and Paste
Don't forget to specify the reference point when cutting or copying, else nothing happens. Keep LC opened during the process, you can have several drawings opened at the same time, preferably using Drawings - Tab mode.
You first have to select the entities to be copied, in case this is not obvious.
Edit: or did you want to copy a drawing from within Draftsight into LibreCAD via the Windows clipboard? No, that's not possible.
If LibreCAD can't open the Draftsight DWG's (DWG import is to be considered experimental only), it's best to use an external tool to convert them to DXF, or open and resave them in Draftsight to DXF as long as it is still working.
The most recent version LibreCAD can read is dxf 2007.