“We had 3,000 people outside the center the other day,” he said.

The migrants and refugees land at all hours, packed into inflatable boats that should hold 15, according to the manufacturer’s instructions stamped clearly on the side of boats. But they usually hold 40, sometimes more. They cross from Turkey, where they have paid smugglers about $1,200 for a place on the boat, more if they want life jackets.

The distance is as little as three and a half miles in some places. But the overloaded boats, taking in water because they sit so low in the sea, can take hours to make the crossings. Passengers that arrive in the night are often exhausted and freezing. Others arrive sunburned. Some end up throwing everything they own overboard, even wheelchairs.

Still, the volunteers who watch for the boats from cliffs say that many of the passengers fall to their knees with happiness when they make it to the rocky beaches here.

It was that way for Rosh A., a 32-year-old Syrian teacher, who asked not to be identified by her last name for fear of what might happen to her family back home on the outskirts of Damascus. Rosh said she had made the trip in less than 24 hours, flying to Beirut, Lebanon, and to Istanbul before climbing into an inflatable boat with her two children and three friends. In Damascus, she said, the bombs arrived regularly and basic services were gone.

“I was dying there every day,” she said, as one of her traveling companions used his smartphone to show a video of explosions and fires erupting in the suburbs of the city. “Yes, it was frightening in that boat, but when I got in it I had a future again.”

Once ashore, however, the group faced a 30-mile walk to register with the authorities. The roads are filled day and night with refugees and migrants trudging toward the port town of Mitilini. Some, like Mr. Nawrozi’s wife, get so exhausted carrying their children that they abandon their belongings along the road.

It is a measure of how few official services there are that those who are rescued by the Coast Guard in the north of the island are brought to Melinda McRostie, who, with her husband, runs a restaurant called the Captain’s Table in the nearby town of Molyvos.