Caveat for all of this: Libraries should be using the Plex Movie scanner & agent. Unknown what will happen if using the Plex Movie (Legacy) agent.
If you manually choose a poster for a movie, it should not change if you rescan or refresh the metadata for that library.
Some users are saying this is not always true. Plex is investigating, but cannot reproduce the problem. See Stop. Changing. My. Posters .
See Move Media Content to a New Location.
Basically, copy the files to the new location and scan to pick them up before removing them from the old location. Read the doc, as there are some additional steps.
This should keep poster selections and any other customizations you’ve made (collections, etc.).
EDIT: I just tested this with my server. It did not keep the same manually selected movie poster. The poster in the new library is the default chosen by Plex.