It is the full faith and credit clause. Otherwise you would need to obtain a driver’s license whenever you transit a state. I am sure that if California thought that they could get away with it they would require a transit permit. They already force people who drive in California a certain amount of time to obtain a California license plate. Yes, if you live in Arizona, which requires an Arizona license for vehicle owned by residents and used on it’s roads, and work and drive in California a certain amount, you must obtain a California license plate; but not a driver’s license. Even California won’t try that. So I can see this being appropriate.
A few states don't require any permit for carrying arms any way one pleases. There's no "thing" (compared with drivers license & marriage/birth record) to give full faith and credit to; for people who live in VT, AK, AZ and WY.
Getting the feds involved will lead to trouble. The federal government is as dishonest and untrustworthy as government can get, when it comes to the RKBA.