Regrettably, G-r's analysis of the full faith and credit clause is not quite comprehensive.
Marriages, which involve a status change recognized by all 50 states and the federal government, fall within the purview of full faith and credit, as do divorces, birth and death certificates, etc. These all involve what without court intervention or a once in a lifetime event, is a permanent status change.
Other permits and licenses, which allow the holder to perform a regulated activity, restrict the permittee to a geographical area, a predetermined period of time, or both. i.e. Plumbers and Lawyers can pursue their trades only in the areas issuing their licenses and must renew them regularly. Heck, if you move into many areas, you even have to re-license your dog!
Driver's licenses, vehicle registrations, and other mutli-state recognized permits are so recognized by states' legislation, states' agreements, or congressional mandate. Even so, when relocating, one's prior license is valid only for a very limited time.
Imagine the confusion that would ensue if a permit to build a structure in one state was valid when the contractor decided to construct the building in another state!
That said, I believe that my carry permit was given to me by my creator, and recognized by my government in the bill of rights!