My wife's Dodge is keyless. If the fob battery gets low a warning message pops up on dashboard. If the battery goes completely kaput the door opens with a key and the car can be started by holding the fob against the dash. Someplace, I forget where. The car can then be started. Don't know if the fob is moved far enough away it stops, we haven't had the problem yet.
I don't think the car will just shutdown if it loses fob connection. I doubt the manufacturer's legal department would find that a good idea.
-Tom