Amro Akoush and his family fled their home in northwest Syria with no time to pack a bag and no vehicle to escape the machine-gun fire and falling bombs. Mr. Akoush carried only a son and daughter, one in each arm.

The family walked along darkened streets to the outskirts of Atarib and hid in an olive grove, waiting anxiously for a friend to ferry them away from the barrage. “It took about 40 minutes,” Mr. Akoush said of their escape on foot, “but it felt like a year.”

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