Greek authorities were transferring around 570 migrants from an overcrowded camp on the island of Lesbos on Sunday, officials said.
A regularly scheduled ferry left Lesbos carrying the migrants, who used to live in the migrant camp of Moria.
Authorities say this is part of the plan to reduce overcrowding at the camp, where about 13,000 people live in a space designed for 3,000.
Migrants, most of them Afghans, have protested, sometimes violently, over the prevailing conditions, demanding their transfer out of Moria and a quick response to their asylum demands.
Authorities say the 570 are among what they term "vulnerable categories" — families, single women with children and unaccompanied minors.
The ferry will arrive at Piraeus, the main port in Athens, on Monday morning. The migrants will be moved to a camp near the northern city of Thessaloniki.
Separately, Greek authorities said a Syrian toddler drowned at a beach in southern Greece after the boy wandered away from his parents and exited a migrant camp.
The Ministry of Citizen Protection, which oversees Greek police, says that the boy was 2 years old. The boy drowned Sunday afternoon after leaving the camp in the city of Andravida in the western Peloponnese region. An inquiry has been launched by Greek officials.
Germany's top security official, meanwhile, said European Union nations need to work better together on the issue of migration or they risk facing a new flood of asylum-seekers that could rival that of 2015.
Following a trip last week to Turkey and Greece, Interior Minister Horst Seehofer told the Bild newspaper on Sunday that "we need to do more to help our European partners with the controls on the EU's external borders."
He said "if we don't do that, we will experience a surge of refugees like 2015, perhaps even bigger."
Germany alone took in 890,000 migrants in 2015.
The comments come ahead of meetings Tuesday with EU interior ministers on the issue.
Seehofer also tells the Welt newspaper that Europe needs to do more to help Turkey in dealing with millions of Syrian refugees.