After Chancellor Angela Merkel said last year that Germany would welcome refugees, boats started swarming in by the thousands. The Greek government, in the midst of an economic and political crisis, was woefully unprepared. So the village fishermen sprang into action, racing toward waterlogged dinghies as screams echoed over the water.

“Our people were in shock — there were so many babies,” Mr. Valamios recalled. “We took the babies first, then returned for the adults. Often you didn’t know if the children would wind up orphans.”

He paused, then clenched his jaw. “We saw many people die.”

The village soon set up a rescue system. If someone saw a migrant boat in trouble, he or she would alert the fishermen to head out. Residents gathered on shore to meet incoming boats and help survivors, who at one point numbered around 5,000 a day. Women, led by village grandmothers, took the newcomers to a small house, where they dressed them in donated clothes and administered milk to babies.