Per Title 1, Section 926A of the 1968 Gun Control Act, it is only
federally protected as legal interstate transport if the person is legally allowed to carry / posses the firearm at both the origin and destination, and is possessing / carrying for lawful purposes:
Sec. 926A. Interstate transportation of firearms
Notwithstanding any other provision of any law or any rule or regulation of a State or any political subdivision thereof, any person who is not otherwise prohibited by this chapter from transporting, shipping, or receiving a firearm shall be entitled to transport a firearm for any lawful purpose from any place where he may lawfully possess and carry such firearm to any other place where he may lawfully possess and carry such firearm if, during such transportation the firearm is unloaded, and neither the firearm nor any ammunition being transported is readily accessible or is directly accessible from the passenger compartment of such transporting vehicle:Provided, That in the case of a vehicle without a compartment separate from the driver's compartment the firearm or ammunition shall be contained in a locked container other than the glove compartment or console.
Federal law restricts FFL sales of handguns to persons under 21 years of age, and possession of any firearm to those under 18. From what I have found it neither prohibits nor explicitly allows for private sales and/or possession to persons under 21 of a handgun. So the state/local laws should govern there. To be OK federally under T1: SEC926, it seems to me that you must verify the above in both the starting and ending jurisdictions of your interstate move.