For a change of address you need the same paperwork as the first day you walk in there except for the fingerprints... Big, long form with all your detailed ID, questions on "Have you ever.... are you under court order for.....domestic, mental, felony, member of an organization...blah blah blah". It also requires two references who must answer letters and forms from the PD inquiring as to your fitness to own a gun, etc. etc. Finish that and stick an $18 money order to it for NJ to do your background check. Then there's the mental health check permission form giving them permission to release any mental health records to the PD. Then you wait for all that to happen.
I saved time by ordering 3 pistol purchase permits or I would have to wait that long AGAIN for them. They will piggyback them. Those are $2 each and I can only buy one handgun per month and the permits are only good for 90 days, but can be renewed for 90 more, but only once.
State police run a background check (NICS I guess) and collect all the money while local PD does all the legwork. Takes 3-5 weeks and I'll have a new paper card (with no photo) with my new address on it. But they know I'm tenacious....