Hani Aziz, a refugee specialist who is an Iraqi refugee himself, was assigned to pick up the family of three at 8 p.m., at the end of a long journey flying from Jordan to Frankfurt to Chicago and, finally, to Detroit.

Two days earlier, Mr. Snyder had proclaimed his opposition to new Syrian refugees entering the state. The specialists mentioned his name defiantly.

“If Snyder’s at the airport tomorrow, pushing them back onto the plane, then we know he’s for real,” Sean de Four, the vice president of children and family services at the agency, said wryly.

But when Wednesday evening came, the family’s flight was delayed for almost four hours. Standing in the arrivals terminal beneath an enormous Christmas wreath, Mr. Aziz scanned the crowds nervously, not knowing anything about the family except for names.

Finally, just after midnight, the family emerged, looking remarkably unrumpled: Nayef Buteh, 45; his wife, Feryal Jabur, 41; and their 8-year-old son, Arab.

Ms. Jabur was poised and elegant but sank onto a bench near the baggage claim.

“It was very tiring,” she said through a translator, looking glassy-eyed and exhausted. The couple’s son, wearing a black bomber jacket and jeans, slumped wordlessly next to her and lowered his dark eyelashes.

Mr. Buteh was polite but agitated, his eyes darting toward the exit. It had been 10 hours since his last cigarette. He stepped out into the mild November air and lit up.