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The cause of their deaths is unknown, but severe crowding can lead to crush injuries, heart attacks and suffocation.
“Conditions on the ground remain extremely challenging, but we are doing everything we can to manage the situation as safely and securely as possible,” the UK Defence Ministry said on Sunday.
At least 12 people were killed earlier in and near the single-runway airfield, Nato and Taliban officials have said.
Some were shot and others died in stampedes, said witnesses.
Britain's opposition Labour Party said hundreds of people had been shot at, beaten or turned back while trying to reach the airport.
Shadow Foreign Secretary Lisa Nandy said Afghan aid workers were put in a perilous situation by having to show documentation at Taliban checkpoints which linked them to Nato forces.
"We continue to deal with significant numbers who are unable to reach the airport due to checkpoints or other challenges, including massed crowds seeking access," she said.
On Saturday, the American embassy told citizens not to travel to the airport without instruction from a US government representative.
Fears of a renewed threat from the ISIS affiliate in Afghanistan prompted US military planes to perform rapid, diving combat landings at the airport, surrounded by Taliban fighters.
Other aircraft set off flares on take-off to confuse any heat-seeking missiles.
Officials described the ISIS threat as significant. The militants have battled the Taliban in the past.
No ISIS attacks have been confirmed since the Taliban retook control of Afghanistan.
The Taliban made people queue outside the airport's main gates and banned crowding along its perimeter, witnesses said.
The witnesses said there was no violence or confusion at the airport as dawn broke on Sunday, although long queues were forming.
The Taliban's swift takeover of Afghanistan has sparked fears of reprisals and a return to the harsh interpretation of Islamic law the group enforced during its previous reign two decades ago.
Crowds have grown at the airport in the heat and dust over the past week, hindering rescue missions for thousands of diplomats, other foreign civilians and Afghans who worked with western organisations in the country.
Australia ran four evacuation flights in and out of Kabul on Saturday night, taking more than 300 people, its prime minister, Scott Morrison, said. Alongside the country's own citizens, passengers included Afghans with Australian visas alongside New Zealanders, Britons and Americans.
Switzerland postponed a charter flight from Kabul on Saturday because of the chaos at the airport.
In a Pentagon briefing, US Army Maj Gen William Taylor said 5,800 American troops remained at the airport and that it “remains secure”.
He said his country had flown out 17,000 people, including 2,500 Americans, from Kabul in the past week.
The White House said US President Joe Biden would provide an update on Sunday on the evacuation mission.
He is due to speak after meeting his national security team to hear intelligence, security and diplomatic updates on the evolving situation in Afghanistan, Washington said.
Taliban leaders are trying to hammer out a new government after retaking the country.
Their forces swept across the country when US-led forces withdrew and Afghanistan's western-backed government and military crumbled.
Mr Biden has come in for severe criticism over the situation. His predecessor Donald Trump called it “the greatest foreign policy humiliation” in US history, even though his own administration negotiated the pull-out that prompted the collapse.
“Biden's botched exit from Afghanistan is the most astonishing display of gross incompetence by a nation's leader, perhaps at any time,” Mr Trump said at a rally in Alabama.
In Qatar, which is hosting thousands of evacuees until they can enter a third country, Afghans who fled described despair at leaving loved ones behind and fear for their own uncertain future.
A law student spoke of looting by the Taliban as they took control of Kabul, and of militants intimidating people who travelled to the airport. He left behind his bride, whom he married in a video call before leaving.
“Our minds are back home because our families remain,” he said, on condition of anonymity out of concern for relatives left behind.
The Taliban's co-founder, Abdul Ghani Baradar, arrived in the Afghan capital on Saturday. He is due to hold talks with other leaders to prepare a new model for government, which a Taliban official said would take shape over the next few weeks.
“Experts from the former government will be brought in for crisis management,” and it would have teams to tackle internal security and financial concerns, the official said.
When in power from 1996 to 2001, the Taliban barred women from leaving home alone. They had to be shrouded in burqas and chaperoned by male relatives, and girls were forbidden from attending school.
The movement has sought to present a more moderate face since returning to power, saying it wants peace and will respect the rights of women within the framework of Islamic law.