What happens if a luggage gets caught between the doors of train as it closes?

Depend on the Train, normally a dispatcher checks the doors are closed before departure.
Safety feature in the train stops it from moving if doors are not fully closed.
Some metro systems have sensors in the path of the door stopping doors from closing if anything is in the way.

you can see this a lot on the London underground when they are crowded and a passengers backpack is in the way.