"Strangulation" seems to me an over-dramatic way of putting it. "Starvation" would be better. The supply of fresh hydrogen stops, so new stars stop being created. The old stars continue burning, some go supernova and blast out metals.Like when you stop adding wood to a fire, the logs already on the fire continue burning and the amount of ash increases.
All this suggests that there is not an indefinite supply of intergalactic hydrogen, so once the galaxy has pulled in all the hydrogen in its immediate vicinity, it will slowly starve.