But terrified Afghans continue to try to flee, deepening a tragedy at Kabul airport where the United States and its allies have been unable to cope with the huge numbers of people trying to get on evacuation flights.
Britain's defence ministry said Sunday seven people had died in the crowds, without giving further details.
A journalist, who was among a group of other media workers and academics lucky enough to get to the airport on Sunday for a flight, described desperate scenes of people surrounding their bus on the way in.
"They were showing us their passports and shouting 'take us with you... please take us with you'," the journalist told AFP.
"The Taliban fighter in the truck ahead of us had to shoot in the air to make them go away."
Britain's Sky News on Saturday aired footage of at least three bodies covered in white tarpaulin outside the airport. It was not clear how they had died.
Sky reporter Stuart Ramsay, who was at the airport, called the deaths "inevitable" and said people were being "crushed", while others were "dehydrated and terrified".
The footage was the latest image of utter despair, after video of a baby being lifted over a wall at the airport and horror scenes of people hanging onto departing planes.
The United States, which has thousands of troops trying to secure the airport, has set a deadline to complete the evacuations by 31 August.
But there are up to 15,000 Americans and 50,000 to 60,000 Afghan allies who need to be evacuated, according to the Biden administration.
There are countless others who fear repression under the Taliban and are also trying to flee.
US President Joe Biden has described the evacuation operations as "one of the largest, most difficult airlifts in history".
The situation was further complicated on Saturday when the US government warned its citizens to stay away from the airport because of "security threats".
No specific reason was given, but a White House official later said Biden had been briefed on security threats, including from the Islamic State jihadist group.
EU foreign policy chief Josep Borrell gave a bleak assessment of whether the airlift would succeed.
"They want to evacuate 60,000 people between now and the end of this month. It's mathematically impossible," he told AFP.
Borrell added that "we have complained" to the Americans that their airport security was overly strict and hampering attempts by Afghans who worked for the Europeans to enter.
On Saturday, the Pentagon said 17,000 people had been taken out since the operation began on 14 August, including 2,500 Americans.
Thousands more have left on other foreign military flights.
The Taliban have been publicly content to allow the US military oversee the airlift, while focusing on how they will run the country once the foreign forces leave.
Taliban co-founder Mullah Abdul Ghani Baradar flew into Kabul and planned to meet jihadi leaders, elders and politicians in the coming days, an official told AFP.
Among them are leaders of the Haqqani network, a US-designated terrorist organisation with million-dollar bounties on its leadership.
The Taliban stunned the world when they swept into Kabul last week, ending two decades of war, facing virtually no opposition from government forces that had been trained and equipped by the US-led alliance.
However, there have been since been flickers of resistance with some ex-government troops gathering in the Panjshir Valley, a mountainous region north of Kabul long known as an anti-Taliban bastion.
One of the leaders of the movement, named the National Resistance Front, is the son of famed anti-Taliban commander Ahmad Shah Massoud.
The NRF is prepared for a "long-term conflict" but is also still seeking to negotiate with the Taliban about an inclusive government, its spokesman told AFP in an interview.
"The conditions for a peace deal with the Taliban are decentralisation, a system that ensures social justice, equality, rights, and freedom for all," Ali Maisam Nazary said.