Ok, I learn something every day. If I have my key fob in my pocket, am driving the vehicle, jump out .... what happens? Does the vehicle shut down?
On many of the newer vehicles, yes. This is mainly the push-start ones (the ones with a keyfob that has no metal key with teeth that has to be slid into an ignition switch). These are the keyfobs that allow you to start the car by pushing a button as long as the keyfob is somewhere in the car (i.e. I start mine while my keyfob stays in my pocket). IIRC some of the earlier versions of cars with these keyfobs would not shut off if the keyfob leaves the vehicle (need the keyfob to start, but don't need it to stay running). Newer ones need the keyfob to stay in or near the car to keep running.
Shouldn't, because most cars are capable of being driven, even if the fob's battery dies. In fact, most cars can start with a dead fob by placing the fob in a particular spot on the dash. This is something everyone should know how to do just in case. However, if you power the car off, and jump out and run away, the car can't be restarted if you have the fob with you.