Evacuations have gone on for more than a week as government forces have retaken the vast majority of eastern Aleppo, which rebels held for more than four years after the Arab Spring uprising.
Emptying the enclave would pave the way for the regime to take all of Aleppo in a major turning point in Syria's brutal civil war that has raged for almost six years and left some 400,000 people dead.
Syrian regime forces as well as pro-Assad militia entered eastern Aleppo by ground in late November, backed by government airstrikes. The regime and Russia, its most powerful ally, have decimated parts of it with strikes.
More than 4,000 fighters were taken in private cars, vans and pickup trucks to rural areas in Aleppo province on Wednesday and Thursday, said Krista Armstrong, a spokeswoman for the ICRC's Middle East operations, but thousands still remain.
"The bad weather, including heavy snow and wind, and the poor state of vehicles mean things are moving much more slowly than expected," Armstrong said, adding that many vehicles had to be towed during the evacuations.
An activist with the Aleppo Media Center who arrived in the city of Idlib on Thursday morning told CNN that his bus was stuck at a crossing for almost 48 hours under severe weather conditions.
Armstrong said the evacuations would continue through the night and likely into Friday.
The ICRC has said that all sick, wounded and other vulnerable people, who were given priority, have been evacuated and that eastern Aleppo's last hospital was empty.
Around 34,000 people have so far been evacuated, Armstrong said.
The evacuations come under a complex people-swap agreement that essentially allows rebels to go to rebel-held areas and pro-regime civilians and fighters to be moved to regime-controlled places.
As people leave eastern Aleppo, other civilians are transferred to the enclave from the cities of Al-Foua and Kefraya in Idlib province under the deal.
A 'devastating' winter
The snow has brought other challenges to humanitarian organizations dealing with the exodus.
The Norwegian Refugee Council, which set up a displacement camp north of Aleppo, is using kerosene lamps and blankets to keep people warm and free from serious illness in these temporary shelters.
"Our priority is getting people somewhere dry and warm because the effects of winter could be devastating, not just in Aleppo but in other parts of the country where people are displaced as well," Thomas White, the council's Syria response director, told CNN.
"In some areas, organizations like ours are distributing heaters. Otherwise essentially it's firewood. And for people in other parts of the country, buying fuel for heating is just not affordable."
He said many people in eastern Aleppo were still sleeping in bombed out or abandoned buildings.
Images show the tops of tent cities blanketed in snow as the displaced make long journeys by foot with all their belongings. Others cart food on their heads along muddy paved roads amid the snowfall.
But the weather hasn't dampened children's spirits -- some were photographed in camps enjoying a snow fight, and pictures of snowmen in Idlib province have been shared on social media.
Such images of joy are rare in Syria's conflict.
But Save the Children said Wednesday that while the evacuations may bring some reprieve from the conflict, they were not cause for celebration. Many of the displaced are going from one war zone to the another.
"These families have been forced from their homes after unimaginable suffering and are now staying in tents and abandoned buildings in a snowstorm, in an area which was already overwhelmed with displaced people," said Nick Finney, Save the Children's northwest Syria country director.
Dr. Mounir Hakimi, chairman of Save the Children's partner Syria Relief, said 30 severely injured children were taken to a charity-backed hospital over three days this week
"Many of the cases we're seeing are of infections from wounds, which couldn't be treated properly under siege, so often we're having to amputate limbs that can no longer be saved," Hakimi said.
"At the reception points, people coming off the buses were starving and exhausted -- I never thought people would be so desperate just to eat a biscuit. We need more blankets and food supplies, but the snow is also going to make our work very difficult in the coming days."