I think you must make a distinction between carrying/discharging a firearm and simply transporting a firearm. I would have no reservations transporting a cased, unloaded .22 rifle while traveling.
Once you reach your destination, state and local law will govern firearm possession but here just use common sense. Don't discharging or brandish your firearm where it's illegal or you're unsure of the law. Display or shoot only where you are certain it's lawful and you have little to fear.
TRANSPORTATION BY MOTOR VEHICLE
In most states, firearms may be transported legally if they are unloaded, cased, and locked in the automobile trunk or otherwise inaccessible to the driver or any passenger. The exceptions to this rule apply mainly to transportation of handguns and so-called “assault weapons.” The myriad and conflicting legal requirements for firearm transportation through the states make caution the key for travelers of which you must consult local law.
If you travel with a trailer or camper that is hauled by an automobile, it is advisable to transport the firearms unloaded, cased and locked in the trunk of the car. If your vehicle is of the type in which driving and living spaces are not separated, the problem becomes one of access. If the firearm is carried on or about the person, or placed in the camper where it is readily accessible to the driver or any passenger, state and local laws regarding concealed carrying of firearms may apply. It is recommended, therefore, that the firearm be transported unloaded, cased, and placed in a locked rear compartment of the camper or mobile home, where it is inaccessible to the driver or any passenger.
Once you reach your destination, state and local law will govern the ownership, possession, and transportation of your firearms.
Source NRA: https://www.nraila.org/articles/2015...transportation