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 don’t 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 don’t fit in the Panama Canal. I don’t know I would want my car on the deck in the “howling 70s”.