The leaders agreed to establish voluntary screening centers on European soil, to ease the burden on countries like Italy, Spain and Greece where migrants first arrive and are registered. They also agreed to study setting up similar centers outside Europe, in North Africa, for example, to screen migrants before they arrive.

Ms. Merkel had even more at stake than the Italians, under pressure from within her own government to solve the problem of migrants coming into Germany after having registered in other countries. The Italian concern has been to stop migrants from coming to Europe in the first place.

Inside Europe, the proposed centers would house migrants until they are screened, with the idea of deciding their fate more efficiently and sending back those who do not qualify as refugees. But no country has so far volunteered to host such a center.

Outside Europe, centers would be designed to reduce the number of migrants who risk the sea voyage to the Continent and to disrupt the black economy of people-smuggling. Those rescued at sea could be returned to those centers for screening, not brought to Europe. But again, it was not clear which African countries might agree to house such platforms, or whether they would be compatible with international law.

Addressing the concerns in Germany about registered migrants moving within Europe to try to settle there, the European leaders simply promised to “take all necessary internal legislative and administrative measures to counter such movements.” The intent is to prevent the setting up of internal borders within the Schengen free-travel area, which could destroy the principle of borderless movement of people and goods.