Policies put in place by a number of countries along the path from Greece to Germany, the preferred destination for most migrants, have effectively limited passage across the borders to Syrians and Iraqis while blocking the progress of Afghans and people from a number of Middle Eastern countries and North Africa.

On Thursday, 200 people occupied the train tracks on the border near Idomeni, holding signs saying “Merkel, Help Us” and “We Just Want to Pass.” Aid agencies warned of a possible risk of infectious diseases, given the area’s limited resources.

Mr. Tusk’s visit to Greece was his second in just over two weeks. Greece, still recovering from an economic meltdown in 2009 that led to three bailout packages from Europe, is bracing for the migrant crisis to continue for the next two to three years, Yiannis Mouzalas, a minister for migration policy, told a group of Greek mayors this week.

Over the past few days, Mr. Tusk, a former prime minister of Poland, has made a sort of reverse pilgrimage along the so-called Western Balkans migration route, visiting Austria, Slovenia, Croatia and Macedonia and Greece before reaching Turkey. “Greece and the Greek people are paying a very high price for the problem they themselves did not create,” Mr. Tusk said. “I want to state here very clearly that the European Union will not leave Greece alone.”

Mr. Tusk’s talks in Greece and Turkey were a prelude to a meeting in Brussels on Monday of European Union and Turkish leaders to address the crisis. In November, the European Union offered organizations helping refugees in Turkey 3 billion euros, or about $3.3 billion, to help deal with the situation. But Turkey — which is itself sheltering 2.7 million of an estimated 4.8 million Syrian refugees — is asking for more support for its position in dealing with the civil war in Syria and other regional issues.