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WATCH: State Department’s Ned Price says U.S. struggling to speed Kabul airlift amid hurdles

The United States struggled Thursday to pick up the pace of American and Afghan evacuations at Kabul airport, constrained by obstacles ranging from armed Taliban checkpoints to paperwork problems.

Watch Price’s remarks in the player above.

With an Aug. 31 deadline looming, tens of thousands remained to be airlifted from the chaotic country.

Taliban fighters and their checkpoints ringed the airport  – major barriers for Afghans who fear that their past work with Westerners makes them prime targets for retribution. Hundreds of Afghans who lacked any papers or promises of flights also congregated at the airport, adding to the chaos. It didn’t help that many of the Taliban fighters could not read the documents.

In one hopeful sign, State Department spokesman Ned Price said in Washington that 6,000 people were cleared for evacuation Thursday and were expected to board military flights in coming hours. That would mark a major increase from recent days. About 2,000 passengers were flown out on each of the past two days, Pentagon spokesman John Kirby said.

At the current rate it would be difficult for the U.S. to evacuate all of the Americans and Afghans who are qualified for and seeking evacuation by Aug. 31.

President Joe  Biden said Wednesday he would ensure no American was left behind, even if that meant staying beyond August, an arbitrary deadline that he set weeks before the Taliban climaxed a stunning military victory by taking Kabul last weekend. It was not clear if Biden might consider extending the deadline for evacuees who aren’t American citizens.

At the airport, military evacuation flights continued, but access remained difficult for many. On Thursday, Taliban militants fired into the air to try to control the crowds gathered at the airport’s blast walls.

Men, women and children fled. U.S. Navy fighter jets flew overhead, a standard military precaution but also a reminder to the Taliban that the U.S. has firepower to respond to a combat crisis.

There is no accurate figure of the number of people – Americans, Afghans or others – who are in need of evacuation as the process is almost entirely self-selecting.