I watched a documentary on a carrier once, and they had a custom where it would stop in Hawaii...and I believe family members could come on board and stay on the carrier on its trip to San Diego...details are hazy, but they didn’t seem too concerned about the liability of having civilians on-board. So cars probably aren’t a problem either.
In the Army, we would occasionally put our cars in the motor pool, if we were doing a long training exercise, just to keep them from being broken into.
I dont know if the still do Tiger Cruises but it was a thing as recently as 20 years ago. Before 911. It occurs to me that for a carrier to relocate from Atlantic to Pacific they would have to go around the Horn as they dont fit in the Panama Canal. I dont know I would want my car on the deck in the howling 70s.