There was no violence or confusion at the airport as dawn broke on Sunday, said the witnesses. Although it was early, there were long lines forming, they said.
Australia ran four flights into Kabul on Saturday night, evacuating more than 300 people, including Australians, Afghan visa holders, New Zealanders, U.S. and British citizens, Prime Minister Scott Morrison said.
On Saturday, the United States and Germany told their citizens in Afghanistan to avoid travelling to Kabul airport, citing security risks as thousands of desperate people gathered trying to flee.
At least 12 people have been killed in and around the single-runway airfield since last Sunday, NATO and Taliban officials said. Some were shot and others died in stampedes, witnesses have said.
The Taliban's swift takeover of Afghanistan has sparked fear of reprisals and a return to a harsh version of Islamic law the Sunni Muslim group exercised when it was in power two decades ago.
Crowds have grown at the airport in the heat and dust of the day over the past week, hindering operations as the United States and other nations attempt to evacuate thousands of their diplomats and civilians as well as numerous Afghans. Mothers, fathers and children have pushed up against concrete blast walls in the crush as they seek to get a flight out.
Switzerland postponed a charter flight from Kabul on Saturday because of the chaos at the airport.
Army Major General William Taylor, with the U.S. military's Joint Staff, told a Pentagon briefing that 5,800 U.S. troops remain at the airport and that the facility "remains secure". Taylor said some gates into the airport were temporarily closed and reopened over the past day to facilitate a safe influx of evacuees.
A Taliban official, speaking to Reuters on Saturday, said security risks could not be ruled out but that the group was "aiming to improve the situation and provide a smooth exit" for people trying to leave over the weekend.
Taylor said the United States in the past week has evacuated 17,000 people, including 2,500 Americans, from Kabul. He said in the past day 3,800 people were evacuated on U.S. military and chartered flights.
President Joe Biden will provide an update on Sunday on the evacuation of American citizens and refugees from Afghanistan, the White House said.
The president is to speak at 4 p.m. EDT (2000 GMT), after meeting with his national security team to hear intelligence, security and diplomatic updates on the evolving situation in Afghanistan, the White House said.
Taliban leaders are trying to hammer out a new government after their forces swept across the country as U.S.-led forces pulled out after two decades, with the Western-backed government and military crumbling
Biden has come in for severe criticism over the situation in Afghanistan, including from former President Donald Trump, who called it "the greatest foreign policy humiliation" in U.S. history, even though Trump's administration had negotiated the withdrawal that triggered the collapse. read more
"Biden's botched exit from Afghanistan is the most astonishing display of gross incompetence by a nation's leader, perhaps at any time," Trump told a boisterous rally in Alabama.
In Qatar, which is hosting thousands of evacuees until they can enter a third country, Afghans who fled described in interviews with Reuters despair at leaving behind loved ones while facing their own uncertain future.
A law student spoke of looting by the Taliban as they took control of Kabul, with armed militants intimidating people going to the airport. He left behind his wife, whom he married in a video call before evacuating.
"Our minds are back home because our families remain," he said on condition of anonymity due to concerns for relatives left behind.
The Taliban's co-founder, Mullah Baradar, has arrived in the Afghan capital for talks with other leaders.
The Taliban official, speaking on condition of anonymity, said Baradar will hold meetings to prepare a new model for governing Afghanistan in the next few weeks, with separate teams tackling internal security and financial issues.
"Experts from the former government will be brought in for crisis management," the official said.
The Taliban follow an ultra-hardline version of Islam. They have sought to present a more moderate face since returning to power, saying they want peace and will respect the rights of women within the framework of Islamic law.
When in power from 1996-2001, also guided by Islamic law, the Taliban stopped women from working or going out without wearing an all-enveloping burqa and stopped girls from going to school.