What shuts down the magma flow in a volcano?

The magma goes through a vent or various cracks in a volcano. Usually high pressure as a result of intense heat forces magma out through these outlets. When it reaches the surface of the earth, the temperatures are low and immediately the cooling process starts. On cooling the Magma may solidify at the mouth of the vent or crater to form a dome or A volcanic plug. These features obstruct the flow of magma.

The earth, or more precisely, the amount of magma. Beneath the volcano's vent is a magma chamber. That fills up with magma. Once the pressure dies, the chamber can seal off by hardening. It takes quite a while usually though.

its possible that over millions of years palte tectonics shift the crust so that it is no longer over the magma chamber thus cutting it off....if the crust over the magma chamber is thicker than it wont neccessarily create a new volcano. SOme times it does like in the chain of hawiian islands and also the aleutian islands in Alaska