What's the Process for a Galaxy to Die?

Astronomers don't talk about galaxies dying. They use the technical term quenching, which means it stopped forming new stars. But you're right, even if it stopped forming any stars at all, those stars themselves can burn for billions and billions, for sometimes even trillions of years. So even a galaxy that's totally quenched won't make any new stars, but those stars will continue to burn for a very long time.