Forget the Syrian Civil War for a moment. Even without the Sunnis and Shiites competing to give each other machete haircuts every sunny morning, there would still be a permanent Muslim refugee crisis.
The vast majority of civil wars over the last ten years have taken place in Muslim countries. Muslim countries are also some of the poorest in the world. And Muslim countries also have high birth rates.
Combine violence and poverty with a population boom and you get a permanent migration crisis.