The events came following an announcement by Turkey that it would open its border with Europe.
A group of residents of the Greek island of Lesbos tried to prevent migrants from arriving in the country, local media and AFP journalists at the scene have reported.
Furious residents yelling "go back to Turkey" tried to prevent a boat carrying migrants from disembarking on the island.
Migrants tried to cross into Europe in larger numbers after the Turkish government said they would no longer prevent refugees from travelling to Europe after 33 of its troops were killed in northern Syria.
An estimated 13,000 migrants travelled to Turkey's border with Greece, the International Organisation for Migration said in a statement. Some of those who spent the night at the border included families with young children, the organisation said.
Greece is determined to stop migrants attempting to cross their border with Turkey, with the foreign ministry tweeting that "no one can cross the Greek borders".
Some residents turned on journalists present at the Port of Thermi in Lesbos. One journalist for Der Spiegel wrote on Twitter that he was harassed and hit with sticks.
In another incident by the border town of Erdine, Greek police fired tear gas at hundreds of migrants.
Turkey's Minister of foreign affairs posted the photos of border violence, tweeting: "Look who’s lecturing us on international law! They’re shamelessly throwing tear gas bombs on thousands of innocents piled at their gates. We don’t have an obligation to stop people leaving our country but #Greece has the duty to treat them as human beings!"
Watch a report on the situation at the land border in the video player above.
President Recep Tayyip Erdogan said his country already has 3.6 million Syrian refugees to look after and announced that the border to Europe was open.
"Approximately 18,000 have forced their way through the gates but today it could reach 25 to 30,000. And we will not close these doors from now on, and this will continue," he said on Saturday morning.
Journalists and observers posted videos showing migrants loading on to busses to head towards the border and some scenes showed migrants entangled in barbed wire.
The European Commission and Council "have been following closely and with concern, the situation at the EU external borders with Turkey," said commission president Ursula von der Leyen.
"Our top priority at this stage is to ensure that Greece and Bulgaria have our full support. We stand ready to provide additional support including through Frontex on the land border," von der Leyen added.
"A situation like 2015 must never be repeated. Our goal must be to properly protect the EU's external borders, stop illegal migrants there and stop waving," Austrian chancellor Sebastian Kurz tweeted.
The EU came to an agreement with Turkey in 2016 to try to stop "the flow of irregular migration". The EU provided billions of euros in assistance to Turkey as part of the deal.