Non-resident FID cards are required if you want to purchase a gun here in NJ. You NEVER need an ID card for ammo purchases. For pistol ammo, you must show a DL or some other type of ID.
If you move to NJ with firearms you legally owned in your last state of residence, you DO NOT need an FID card. (though, it is recommended that you get one just to avoid any problems in the future)
If you are travelling to an event here in NJ, you are covered by the Federal transport law, which can also be found on the NJSP website. In short, the weapon must be legal where you start and end your trip, unloaded, locked up, etc. So, if you are coming to a shoot in NJ with a ban compliant weapon, you are good to go.
If you are travelling through NJ to another state, same as above except the legality of the weapon is based on whatever the start and end points of the trip are... I.e., going from PA to VA via NJ you could have a super-evil bayo lug/folding stock/baby-killer AK and would be fine.
Of course, that's the way its supposed
to work. You never know what might happen if you get pulled over and some know-nothing academy grad who thinks she just made her career with a gun-running bust...