Yes, it is possible to drive a car without the key fob once the engine is running. You can switch from one park to another without any problem, but the question is how far can you go without a key? The answer lies in the fuel tank - the car won't stop until the fuel runs out. It doesn't matter how far you want to go, it will be impossible to start the vehicle once the ignition is turned off. The process may vary in different vehicles with different technology.
Some key rings do not allow you to cross the area unless you only have it inside the cabin. It all depends on the proximity of the key fob to the vehicle. New cars come with great and extremely fun technology, including keyless ignition and push-button start. But what happens if you lose your key ring? The answer depends on the situation.
For example, you will have a very different experience if you lose your key fob BEFORE you get in the car or after. Simply put, if you lose the remote that allows button start before trying to drive in your car, you simply won't be able to start your vehicle. However, what happens if you lose your command while driving? That's a very different answer, and there are ways to bypass this issue. Most cars are equipped with a means to start your car manually, while others have backups built into the key fob that works quite well without a key.
Therefore, to circumvent such regulation, the car simply stays running so that automakers can argue that the key cannot be removed before the car is put into parking. This would make it trivially easy to violate the regulation if the car cleared the start code as soon as the key fob was too far away, as it would allow a situation where the key can be removed before the car is put into parking. To start a car without a key fob, there is always a backup system placed by several car manufacturers around the direction of the car. Since the car depends on the key fob, you will need to authenticate it before you can put the car in reverse gear or drive. I have heard that there were reports of cars moving away due to drivers not shutting down the engine with the car still running or in reverse.
An important point to understand is that automakers consider that the electronic code transmitted from the key fob to the ignition system is like a physical key, so when you press the button to turn off your car, this code is erased from your start system. The thief uses a radio relay to make your car think that your remote control is nearby, then starts your car and leaves. Older ignition systems require that you place your car in park before removing your key. Some automakers use remote door unlock features that allow you access when locked outside, locks your keys inside or has a disabled key fob.
