The emergency airlift from Kabul hit stumbling blocks on Friday and Saturday as flights were delayed, bottlenecks appeared in reception centres and thousands of Afghans continued to struggle to reach safety.
Here is a round-up of the latest developments:
The United States on Saturday urged its citizens in Afghanistan to avoid travelling to the Kabul airport for now, citing "potential security threats" near its gates.
Most roads in the capital were largely deserted save for the route to the airport, which was choked with people scrambling to join a US-led evacuation.
Conditions outside Hamid Karzai International Airport have been chaotic, with reports of people seeking to leave being beaten by Taliban fighters.
"Because of potential security threats outside the gates at the Kabul airport, we are advising US citizens to avoid travelling to the airport and to avoid airport gates at this time," the alert said.
The US military on Friday said it sent helicopters to rescue more than 150 Americans unable to reach the airport gates, in the first evidence that its forces were willing and able to go beyond the compound to help people seeking evacuation.
At Friday prayers in Kabul, an imam described those trying to flee as not having strong enough religious convictions.
"Those with weak faith are running after or hanging from American planes," he said.
US aircraft flew about 6,000 people, including a couple of hundred US citizens, out of Kabul in the 24 hours to early Friday, until flights had to be halted because of a lack of space at transit bases.
Officials confirmed that evacuation operations stalled for several hours on Friday because a receiving base in Qatar was overflowing and could not take any more evacuees.
Evacuee accounts from Qatar describe sleeping on the floor in sweltering heat in a US aircraft hanger for three days or more, with limited facilities.
More than 7,000 people have been evacuated from Afghanistan to Qatar, an official from the Gulf state said on Saturday.
"At the request of NGOs, educational institutions and international media organisations, we evacuated hundreds of Afghan employees and their families, as well as female students across the country," a Qatari official who declined to be identified told AFP.
That came "in addition to facilitating the evacuation of citizens from the United States of America, Germany and the United Kingdom, among others. Our evacuation effort is ongoing."
Doha will eventually settle up to 8,000 Afghans, according to the Qatari official, who stressed that many of the 7,000 people in Qatar were transiting to third countries.
The US and Germany had agreed on Friday to use the Ramstein US military base in western Germany to ease pressure on Doha.
The base can accommodate about 5,000 people.
Around 1,150 people landed there from Kabul on Saturday, from where they are expected to depart for the US in "a few days", according to a spokeswoman for the base.
Meanwhile, more than 8,500 people have transited the UAE so far, according to the government
Switzerland said the worsening security had delayed an evacuation flight they had organised from the Uzbek capital of Tashkent because too few people had been able to get there from Kabul.
They said the Germans had also postponed evacuation flights from Tashkent.
EU urges members to step up
EU chief Ursula von der Leyen on Saturday appealed to European Union states to take in Afghan refugees flown out from Kabul, promising financial support from Brussels.
"To those who cannot go back or stay home, we have to offer alternatives," said von der Leyen, after visiting a base in northeast Spain that will serve as a reception centre for Afghans who worked for the EU.
"The Commission stands ready to look into the necessary budgetary means to support EU member states who will step up and help resettle refugees," she added.
Compassion amid security fears
Meanwhile, images of US military appearing to aid desperate Afghan parents attracted attention, with one video of a Marine lifting a baby over the razor wire around the airport going viral.
The video, which shows the infant, its nappy slipping off, being pulled up by one arm high above a crowd of Afghans seeking to enter the airport, took over social media nearly one week into the airlift to evacuate foreigners and Afghans from the war-torn country.
"The parent asked the Marines to look after the baby because the baby was ill," Pentagon spokesman John Kirby told reporters.
"The Marine you see reaching over the wall took it to a Norwegian hospital that is at the airport. They treated the child and returned the child to the child's father," he said.
There was no information about the family's fate or status.
The US military released a number of images that appeared to show them caring for Afghan children -- including holding babies and giving water to a young boy in the airport compound.