Has to go thru an FFL at the recv'g end w/a 4473 and NICS check being done.
There's no exemption for out of state gifts to family members or others.
The only exception I'm aware of that exempts the nonresident transfer from going thru an FFL is where a firearm is left to the non-resident as an inheritance in the legal will of the deceased owner.
Make sure that the FFL in NH is willing to accept the firearms from a Non-FFL from out of state for the transfer process if the OP is intending to ship them himself.
Some will,,some will not.
The regs are clear that it is completely legal to do so, but many FFL's will not accept firearms from other than another FFL.
That subject has been beaten to near death already but the real issue here is just to save problems and make sure the transfer goes smoothly.
https://www.atf.gov/firearms/qa/whom-may-unlicensed-person-transfer-firearms-under-gca