I guess it depends is the best answer, I have a similar issue, although I'm not moving around.
Currently I have three addresses, one is my Lat/Long physical location I give to family/friends to visit along with directions, since I'm in unincorporated land, with no "real" address. One is my "imaginary" physical address, for things like bank accounts etc. basically an off milepost marker, but the bank also has my mailing address. The final is my mailing PO Box, my drivers license is addressed as this for instance, although the DMV have my off milepost address too.
Fortunately I live in a state where this kind of thing is not uncommon, and we don't need a concealed carry permit.
However I am a permanent resident, and not transient. Interesting question, you could get a Non-Resident Permit from one of the states that issue them (AZ, TX, UT to name three) for concealed carry, for drivers and vehicle licensing, I guess that would depend on your permanent mailing address, and their regulations.