As I mentioned it is not an easy process.
1. I find that I can make copies of drawings and cad files from the folders themselves. I usually name the copies with something like "ZZZZZ" as the front after the copies have been made (so that way they "fall" to the bottom when sorting a folder).
(e.g. ZZZZZoldname.asm, ZZZZZoldname.dwg)
2. Open the original drawing(s) and from one of the drawing(s) open the generic assembly model.
(e.g. oldname.asm, oldname.dwg)
3. Rename the generic model to the new name chosen. Rename the instances I want to keep (if needed). Rename the drawings (if needed).
(e.g. oldname.asm -> newname.asm, oldname1<oldname>.asm -> newname1<newname>.asm, oldname.dwg->newname.dwg, ....)
4. Delete the instances that are not needed in this family.
(e.g. oldname4<oldname>.asm, oldname5<oldname>.asm)
5. Save the drawings, models, and generic. Then close and erase from memory.
6. In the folder remove the "ZZZZZ"'s from the copies.
7. Go into the "orignal" files and delete the instances now in the new files.
8. Repeat the process as needed.
If the assemblies are a sub componenet, it gets complicated in that you need the models of the parent components open when you are doing the renaming else the connectivity get lost.