The restrictions have left thousands of migrants and refugees stranded in Greece, where that country's prime minister is now calling for a global solution to the crisis.
A boatload of people who have fled their homes across the Mediterranean Sea cried out for help as their vessel floundered off the coast of the Greek island of Kos.
The Turkish coast guard would go on to save them, another disaster averted as more than 350 people have been plucked from the waters over just the past four days.
The number of migrants and refugees arriving in Greece and Italy has risen sharply so far this year.
And UNHCR spokeswoman Stella Nanou says it is severely stretching resources on the ground.
"Since the beginning of January, we've had some 95,000 arrivals in Greece, which is a huge increase in comparison with the same period last year. What we have been urging for a long time is for a common strategy in order to deal with the refugee crisis in Europe. This is not only a Greek problem, so all European states should try to solve this in a common way."
But achieving that solution is proving to be a diplomatic challenge as countries further north on the migration route to Western Europe move to restrict, even close, their borders.
It is leaving a growing bottleneck of people stranded.
Greek prime minister Alexis Tsipras has hit out at unilateral actions by some countries.
"If we want to solve this problem, we have to cooperate closely, coordinate more and share the burden. Greece is part of the possible solution. We will give our aid to these people, because this is our main priority. But at the same time, we will not accept unilateral actions by anyone."
United Nations High Commissioner for Refugees Filippo Grandi has supported his remarks.
Mr Grandi says some countries are breaching both European Union rules and basic rules regarding refugee protection.
"I think that it is very important that all the countries that have reacted in this country-by-country way, in this instinctive way, by closing borders think carefully about it. The response is not closure. It's cooperation."
But at a meeting with Balkan states on the crisis, Austrian foreign minister Sebastian Kurz has defended his country's right to control its borders by limiting the flow of refugees.
"There was not a single attendant today at our meeting who was against a European solution. There is not a single participant who doesn't think it was the best to resolve the situation at the Turkish-Greek border. But we are all aware that, besides working on the European solution, national and regional measures are needed, because, otherwise, there would be an overstrain."
Germany, which has welcomed the migrants and refugees, is starting to feel that strain, too.
It has launched media campaigns now to discourage people from making the trip across the Mediterranean.
Yesterday, it returned 125 migrants to Afghanistan on a chartered flight.
Those aboard said they were going home voluntarily after their bids for asylum were rejected.
Germany says they, like so many others, were victims of unscrupulous people-smugglers.
One Syrian migrant named Lilav had a message for those seeking to exploit their desperation.
"We are human. We are not a prize. We are not money. We are not one, two, three, four, five humans to pay the money."
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