The UN Security Council has voted unanimously to "carry out adequate, neutral monitoring and direct observation" of the evacuations in eastern Aleppo.
The 15-member council demanded all parties in the conflict to provide UN officials and others with a "safe, immediate and unimpeded access" to those who remain in the Syrian city.
Since midnight, some 4,500 people have left the besieged rebel-held enclave of the city to surrounding villages.
The mother of a seven-year-old girl who tweeted about living conditions under the Syrian regime has spoken of her relief to finally reach safety.
Bana al-Abed attracted over 300,000 Twitter followers by providing harrowing updates on her living conditions and near-death experiences when she was growing up in besieged eastern Aleppo.
On Sunday, Bana and her mother Fatemah were some of 12,000 people to be evacuated as part of a deal to return the city to government control.
Holding her arm around her daughter, Fatemah told the activist-run Qasioun News at her relief at leaving, but also the regret of being forced out of her home city and living as a refugee.
We created our Twitter [account] to tell all the world what is happening there [in eastern Aleppo].
We are happy because our voice reached all the world. We are free.
We are feeling - I am sad because I leave my country, I left my soul there. I want to take our freedom there, not to be like a refugee in other countries. I want for my kids a good future. But this is the matter here.
They make us leave our country. We can't stay there because there is a lot of bombs and no clean water, no medicine.
They even targeted hospitals and schools. When we get out, we had a lot of suffering, because we stayed almost 24 hours in a bus without water and food and anything.
We stayed like prisoners, a hostage.
Finally, we arrived here and we thank God and we thank all our friends who have supported us.
Dozens of children who were trapped in an orphanage in the rebel-held enclave of east Aleppo have been evacuated, UNICEF said.
Geert Cappelaere, regional director for the United Nation's Children's Fund, said: "This morning, all 47 children trapped in an orphanage in east Aleppo were evacuated to safety, with some in critical condition from injuries and dehydration."
He said UNICEF and other agencies were assisting in reuniting other evacuated children with their families, and giving them winter clothes and medical care.
Some 12,000 people have been evacuated from east Aleppo in the past few days, according to Turkey's foreign ministry.
The figure includes 4,500 civilians who have managed to leave the city since midnight.
Mevlut Cavusoglu said: "Including 4,500 people in east Aleppo since midnight, 12,000 civilians in total were evacuated so far to the area under control of opposition."
The evacuations are the result of intense negotiations between Turkey - which backs some large rebel groups - and Russia, who support President Bashar al-Assad's regime in Syria.
A seven-year-old girl who gained more than 300,000 followers after tweeting about her ordeal in eastern Aleppo has managed to leave the besieged city, according to an aid organisation.
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Bana al-Abed was included in a convoy of busses that left the city over the past week.
Her departure was confirmed by the head of the Syrian-American Medical Society aid group.
In a tweet on Monday, Ahmad Tarakji said Bana was among "many children" who have arrived in the Aleppo countryside.
A convoy of busses carrying evacuees from eastern Aleppo has arrived at the town of al-Rashideen.
The Syrian Observatory for Human Rights said the new convoy of 21 busses had arrived at the insurgent-held district early on Monday.
Ten buses left the besieged villages of al-Foua and Kefraya near Idlib carrying evacuees through rebel-held territory towards Aleppo on Monday, according to the Syrian Observatory for Human Rights.
The evacuation of wounded and other civilians from the villages is a condition for the evacuation of people in the besieged rebel zone of Aleppo, Syrian state media reported on Sunday.
Tens of thousands of civilians desperate to leave Aleppo were left stranded for 48 hours over the weekend after a ceasefire deal broke down.Read the full story ›
Five buses have been allowed to leave eastern Aleppo to continue on to rebel-held areas after being held for hours, the Syrian Observatory for Human Rights said.
One ambulance was also escorted out of east Aleppo, according to a senior Red Cross official.