I personally think this is a very unrealistic part of the current game. No rocket in this world magically shuts down the previous stage's engine, and neither should the game.
I think it would add a lot more realism and also open up opportunities for a lot of creative staging such as the famous korolev cross used by Soyuz and earlier R7 rockets.
So why am I suggesting this? This is all way too common both in other games as well as in real life. For instance, the first stage of the falcon 1 was not shut down properly on one of it's earlier flights; resulting collision with the 2nd stage, failing the mission.. Moreover, the much recent Soyuz MS-10 abort, where due to the nature of the Soyuz separating with main engines still providing thrust, even the failure of one of four liquid oxygen vent/ejection valves responsible for pushing the booster away from the center stage resulted in the abort of the mission.
I'm suggesting to have previous stage continue to fire unless throttled down by player or the previous stage is out of propellant.