I have two cats, both about 2.5 yrs old and have been a traveler for the past 2 years. Last year the girls did over 12000 miles by car and another 2-3K by air.
When in the car I allow them to roam free, they are both strictly indoor cats. At night they are first into the motel room and in the morning generally the last out.
The only problem I've had is in the morning. Most motels are getting smart and enclosing the base of the beds so animals can't clime up under them. Those that don't may require you to turn the bed over to find kitty. I've found Motel 6 the most accommodating.
Of all the traveling, only once has one of the girls tried and gotten out at a pit stop. I got lucky and she ran into a dead end corner so freedom was short. I initially started out providing water and food during the trip but soon realized they weren't interested. Now I make that a priority when we stop for the day. I also travel with their litter box. Sometimes they use it but mostly one or the other uses it to hide in.
I make it clear to whatever agency/recruiter I'm working with that no cats, no Travel. I haven't had a problem yet. Even if I have them provide housing, they will front any deposit and take it from the 1st pay check. Generally you can find an efficiency motel on your own that doesn't require a deposit and the housekeepers will work with you about the cats and cleaning the room.
Hope this helps!