It’s been a year since Qasim left his home in Sinjar, Iraq. He was fleeing an advance by the Islamic State — the same advance that left tens of thousands of other Yazidis stranded in the mountains, trapped between hunger and dehydration and the threat of mass violence.

Qasim and his parents and three brothers, along with their wives and children, were able to get away from Sinjar, to a . . .