You have to find out local laws in the states you'll be moving the gun through. Do that by contacting law enforcement in those states, some dude got 7 years a few years ago, he did it the kingAddies way. When he was pulled over and asked if he had a firearm he told them yes, and it was illegal in the state he moved to, to take a firearm over state lines.
If you can't in the state you're moving to/through, you'll have to have the weapon transferred to the state you're moving to through an FFL in your current state to an FFL in the state you're moving to.
Or take your chances