Airdrops of food and water — which apparently went awry — were followed by airstrikes, including four on Friday that targeted Islamic State positions around the mountain.

The attacks helped at least some of the Yazidis escape, said Zaim Hassan Harmouch, 66, who said the bombings destroyed the militant positions that had blocked their route out. He led his wife, six sons and seven grandchildren down from the mountain overnight, crossing the border into a Kurdish-held region of Syria and then back into northern Iraq.

“It was because of the planes that we could leave,” he said. “They opened the way.”

Harmouch was among an estimated 50,000 Yazidis who fled the Islamic State assault on Sinjar Aug. 3, after the militants surged into town threatening to kill any of those who did not convert to Islam. He carried his wife on his back all the way, because her knees gave way.