The rebel withdrawal began a month to the day after President Bashar al-Assad's forces launched a new offensive to recapture Aleppo and will hand the regime its biggest victory in more than five years of civil war.
A revived agreement on a ceasefire and the evacuations was announced on Thursday, after an initial plan for civilians and fighters to leave rebel-held parts of east Aleppo collapsed the previous day amid renewed clashes.
The evacuation began with a convoy of ambulances and buses crossing into a government-held district in southern Aleppo around 2:30 pm (1230 GMT).
A Syrian military source told AFP that 951 evacuees, including 108 wounded, were in the convoy. Most were civilians but about 200 rebel fighters were among them, the source said.
The convoy arrived just over an hour later in opposition territory west of the city, a doctor at the scene said.
"Vehicles carrying the wounded have arrived, and the wounded will be transferred to... nearby hospitals for treatment," said Ahmad al-Dbis, who heads a unit of doctors and other volunteers coordinating the evacuation of wounded people.
He spoke to AFP from the transit point near opposition-controlled Khan al-Aassal, about five kilometres (three miles) west of Aleppo city.
'We will return'
The evacuees had spent hours earlier gathering earlier at a staging area in Aleppo's southern Al-Amiriyah district.
An AFP correspondent there saw people piling onto the green buses, filling seats and even sitting on the floor, with some worried that there would not be another chance to evacuate.
Many were in tears and some hesitated to board, afraid they would end up in the hands of regime forces.
On the dusty window of one of the buses someone had written "One day we will return".
Each bus carried a member of the Syrian Red Crescent dressed in the organisation's red uniform, riding at the front next to the driver.
Ingy Sedky, the International Committee of the Red Cross's spokeswoman in Syria, said the first convoy included 13 ambulances and 20 buses carrying civilians.
Once the first convoy arrives safely "it will return and collect more people for a second journey and continue like that. We will go today for as long as conditions allow," she told AFP.
Syrian state television reported that at least 4,000 rebels and their families would be evacuated under the plan.
It said preparations were underway for a second convoy to leave rebel-held territory.
A first evacuation expected to take place on Wednesday morning fell apart, with artillery exchanges and resumed air strikes rocking the city until the early hours of Thursday.
But the agreement, brokered by Syrian regime ally Moscow and opposition supporter Ankara, was revived following fresh talks.
The defence ministry in Moscow said that Syrian authorities had guaranteed the safety of the rebels leaving the city.
The head of the UN-backed humanitarian taskforce for Syria, Jan Egeland, told reporters in Geneva that "most" of those evacuated from Aleppo would head to opposition stronghold Idlib, in Syria's northwest.
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"Russians and others assure us that there will be a pause in the fighting... when we assist the evacuation," Egeland said.
Ambulance 'fired on'
The evacuation was going ahead despite reports earlier Thursday of pro-regime forces firing on an ambulance transporting the injured to Al-Amiriyah, wounding three people including a member of the White Helmets civil defence organisation.
On Wednesday, cold and hungry civilians had gathered for the initial planned evacuation but were instead sent running through the streets searching for cover as fighting resumed.
Russia accused the rebels of having violated the ceasefire while Turkey accused Assad's regime and its supporters of blocking the evacuation.
Iran, another key Assad backer, was reported to have imposed new conditions on the agreement, including the evacuation of some civilians from two Shiite-majority villages in northwestern Syria under rebel siege.
On Thursday, nearly 30 vehicles were headed to Fuaa and Kafraya to evacuate sick and wounded residents, the governor of neighbouring Hama province, Mohamed al-Hazouri, told state news agency SANA.
A Syrian source on the ground told AFP that "1,200 injured and sick people and their families will be evacuated."
Backed by foreign militia forces including fighters from Lebanon's Shiite Hezbollah movement, the advance launched last month made rapid gains, leaving the rebels cornered in a tiny pocket of the territory they had controlled since 2012.
More than 465 civilians, including 62 children, have died in east Aleppo during the assault, according to the Syrian Observatory for Human Rights.
Another 149 civilians, among them 45 children, have been killed by rebel rocket fire on government-held zones in the same period, the Britain-based monitoring group said.
The United Nations and Western countries this week condemned alleged atrocities being carried out by pro-government fighters, including reported summary executions of men, women and children.
A UN panel said on Wednesday that it had also received reports that rebel fighters were blocking civilians from leaving and using them as human shields.
Shrinking rebel territory
More than 310,000 people have been killed since the Syrian conflict began, and over half the population has been displaced, with millions becoming refugees.
The United States and other Western nations, Turkey, and Gulf Arab states all backed opposition forces during the war but their support was limited.
The conflict, which began with anti-government protests that were brutally put down, saw a turning point last year when Russia launched an air war in support of Assad.
With Aleppo out of rebel hands, the largest remaining rebel bastion is Idlib province, which is controlled by an alliance dominated by former Al-Qaeda affiliate Fateh al-Sham Front.
Rebels also hold territory in southern Daraa province and the Ghouta region around Damascus, although the army has been advancing there.
Diplomatic efforts -- including several rounds of peace talks in Geneva -- failed to make headway in resolving the conflict.
After upping its involvement by brokering the Aleppo deal, Turkey said it would meet with Russia and Iran in Moscow on December 27 to discuss a political solution to the entire conflict.
"We are striving to secure a ceasefire throughout the country and for negotiations for a political solution to start," Foreign Minister Mevlut Cavusoglu said.