Some of them made victory signs, shouted "Allahu akbar" (God is greatest) and waved Syrian rebel flags as they advanced towards Ras al Ain.
Turkish President Recep Tayyip Erdogan has said the military action is needed to prevent the creation of a "terror state" along Turkey's border with Syria. The SDF said 22 of its fighters were killed on Wednesday and Thursday.
Children hold pictures, as Syrian Kurd refugees protest at the Ritsona camp, some 80 km north of Athens, on Thursday, after Turkey launched an assault on Kurdish forces in northern Syria with air strikes and explosions reported along the border.
"We have to be aware of this and mobilize the resources of our intelligence to undercut this emerging tangible threat", Putin said.
"President [Recep Tayyip] Erdogan and his regime must face serious consequences for mercilessly attacking our Kurdish allies in northern Syria, who incurred thousands of casualties in the fight against ISIS and helped us protect the homeland", said Cheney.
Turkey has defended its bid to create a "safe zone" free of Kurdish militias which could also house Syrian refugees.
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The SDF have been key allies of the United States in the battle against the Islamic State (IS) group.
"Turkey's operation in northeastern Syria is a result of steps taken by U.S.in this region", Lavrov said. "Turkey knows where I stand".
Le Drian said on France 2 television that the coalition, which includes more than 30 countries, needs to discuss a range of issues because Islamic State could take advantage of the changes on the ground to re-emerge.
U.S. ambassador to the UN Kelly Craft said: "Failure to play by the rules, to protect vulnerable populations, failure to guarantee that IS can not exploit these actions to reconstitute, will have consequences". Inside the town, several mortal shells killed three people, including a nine-month-old Syrian baby.
The divided U.N. Security Council failed to agree on Turkey's offensive in northeast Syria on Thursday, with Europeans demanding a halt to military action and Syrian ally Russian Federation calling for "restraint" and "direct dialogue" between the two countries.
The Ankara-based Anadolu news agency released footage of Turkish armored vehicles crossing the border into northeast Syria late last night, in what was said to be 'the direction of SDF-held territories.' The mobilization prompted local Syrian civilians to flee their towns this morning. The towns of Ras al-Ain and Darbasiya, some 60 km (37 miles) to the east, have been largely deserted as a result of the attack.
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The border strip which USA forces vacated this week, and which is believed to be the focus of Ankara's incursion, is a almost 100-km (62-mile) stretch between the Syrian towns of Tel Abyad and Ras al-Ain. "At the same time, their heavy weapons haven't stopped, they aren't stopping in firing on the village, and civilians have been both wounded and lost their lives", Ahmed said.
Trump has faced rare criticism from senior figures in his Republican Party who accuse him of deserting USA allies.
The Syrian Kurdish fighters, along with USA troops, have been involved in mopping up operations against IS fighters still holed up in the desert after their territorial hold was toppled earlier this year.
But Turkey's Foreign Minister, Mevlut Cavusoglu, dismissed those fears.
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