There, a mass migration, precipitated by increasingly widespread fears about the Sunni militants’ advance, is underway. It has poured about 580,000 refugees into the Kurdistan region, about 200,000 since Monday when ISIS took Sinjar and its surrounding villages, according to Mr. Swanson of the United Nations’ humanitarian assistance office. They are there on top of another 230,000 Syrian refugees.

As ISIS has moved steadily through the disputed areas along the border of Iraqi Kurdistan, civilians have fled into the region.

In village after village, town after town, people were running ahead of rumors that ISIS was coming. The Kurdish forces offered to help people leave, piling them into huge open trucks and handing out water before they set out for the east or west with their tottering loads. Individuals in cars, pickup trucks and farm vehicles with mattresses strapped on with old twine hobbled along the bumpy roads just trying to get away. The old road to Dohuk, which runs across Kurdistan, was filled with cars heading to larger cities.

This most recent exodus has involved primarily the minority Christians, Yazidis, Shabaks, another Muslim sect, and Turkmen Shiites, all of whom the Sunni militants view as heretics, as they do all Shiites.

The particular fear for the Yazidis is that ISIS appears not only to be displacing them and forcing conversions, but also especially targeting them for death.

Until the Yazidi fighters and the Syrian pesh merga opened a route, only small numbers were making it down on their own, relying on their sense of the mountain from years of worshiping on its slopes or in some cases herding sheep and goats there. In some cases, groups of women have come alone, bringing their children while their men stayed on the mountain, some to help the Yazidi fighters.

Mr. Aslan and his family debated with several other families whether to risk going down the mountain. They were not sure how far they would have to walk or whether, when they reached the foot of the mountain, the gunmen they were fleeing would be there, waiting to kill them. When ISIS took the area around Sinjar they cut off routes that would have allowed the Yazidis to go around the mountain and reach the road into Iraqi Kurdistan, effectively driving the Yazidis into the mountain wastes.