US forces running the airport had to stop flights on Monday after thousands of frightened Afghans swamped the facility looking for a flight out.
Flights resumed on Tuesday as the situation came under control.
Britain said it had managed to bring out about 1,000 people a day while Germany flew 130 people out.
France said it had moved out 25 of its nationals and 184 Afghans and Australia said 26 people have arrived on its first flight back from Kabul.
As the Taliban consolidated power, one of their leaders and co-founders, Mullah Abdul Ghani Baradar, returned to Afghanistan for the first time in more than 10 years.
A Taliban official said leaders would show themselves to the world, unlike in the past when they lived in secret.
"Slowly, gradually, the world will see all our leaders, there will be no shadow of secrecy," the senior Taliban official told Reuters.
As Baradar was returning, a Taliban spokesman held the movement's first news briefing since their return to Kabul, suggesting they would impose their laws more softly than during their earlier time in power, between 1996-2001.
"We don't want any internal or external enemies," Zabihullah Mujahid, the Taliban's main spokesman, told reporters.
Women would be allowed to work and study and "will be very active in society but within the framework of Islam", he said.
During their rule, also guided by sharia religious law, the Taliban stopped women from working. Girls were not allowed to go to school and women had to wear all-enveloping burqas to go out and then only when accompanied by a male relative.
Ramiz Alakbarov, UN humanitarian coordinator for Afghanistan, told Reuters in an interview the Taliban had assured the United Nations it can pursue humanitarian work in Afghanistan, which is suffering from a severe drought.
The European Union said it would only cooperate with Taliban authorities if they respected fundamental rights, including those of women.
'TIME WILL TELL'
Many Afghans are sceptical of the Taliban promises. Some said they could only wait and see.
"My family lived under the Taliban and maybe they really want to change or have changed but only time will tell and it's going to become clear very soon," said Ferishta Karimi, who runs a tailoring shop for women.
Mujahid said the Taliban would not seek retribution against former soldiers and government officials, and were granting an amnesty for ex-soldiers as well as contractors and translators who worked for international forces.
"Nobody is going to harm you, nobody is going to knock on your doors," he said, adding that there was a "huge difference" between the Taliban now and 20 years ago.
US President Joe Biden and British Prime Minister Boris Johnson said they had agreed to hold a virtual meeting of Group of Seven leaders next week to discuss a common strategy and approach to Afghanistan.
The decision by Biden, a Democrat, to stick to the withdrawal deal struck last year by his Republican predecessor Donald Trump has stirred widespread criticism at home and among US allies.
Biden said he had to decide between asking US forces to fight endlessly or follow through on Trump's withdrawal deal. He blamed the Taliban takeover on Afghan leaders who fled and the army's reluctance to fight.