how is it arriving into the NH FFL's hands?
Some FFL's will not accept a gun shipped in by a private party.
Anyone can carry one into a shop, that is not a problem. , You can go from MA into NH with a handun no problem no permits, just keep it unloaded in the car, and no CCW loaded
As for shipping only another FFL or manufacturer can use the U S Mail to ship a handgun to a FFL.
All other shipments of handguns have to go "common carrier" which 99% of the time is from a UPS depot, declared, etc etc etc.
Long guns can go by U S Mail by FFL or private sale.
Does this answer the question?