Not being a NJ resident I can only tell you what I have read....
NJ Firearms Purchase ID is for purchasing long guns...doesn't really have anything to do with "keeping" a gun. The FPID is not just for residents. A non NJ resident could obtain one if they wanted to, so you changing your primary residence does not void your FPID. Just keep in mind, if you wanted to change your address in the NJ ID system, you will be going thru the application process all over again.
States don’t have rights. People do.