The Pakistan Army said that the captors of a Canadian-American family held by a Taliban-linked group fled on foot after troops shot at their vehicle's tyres, as it offered a fuller account on Friday of the operation to rescue the hostages.

American Caitlan Coleman and her Canadian husband Joshua Boyle, who were kidnapped while backpacking in Afghanistan in 2012 and had all three of their children in captivity, have left Pakistan after being freed, according to a United States (US) official.

The army said it launched the rescue operation after a tip off from US intelligence that the family had been moved into the tribal areas from across the border in Afghanistan.

Residents in Kurram Agency, where the operation took place, and North Waziristan told AFP they had seen drones flying in the skies above them for several days before the operation.

Director General (DG) Inter-Services Public Relations (ISPR) Maj Gen Asif Ghafoor said Pakistan was told by US intelligence at 4pm Wednesday that the hostages were on the move.

"We sent our troops, traced the vehicle on the basis of intelligence sharing by 1900 hours yesterday (Wednesday) and recovered the hostages," he said in televised comments late Thursday.

Security forces had planned to intercept the vehicle at a checkpoint in Kurram, a security source told AFP ─ but the militants drove it off the road.

The troops tried to stop the vehicle once it had travelled a few miles over the border. "But when the militants refused to halt, they shot out its tyres," Ghafoor told AFP.

The militants "fled on foot", leaving the family in the car, according to Ghafoor, who added that the soldiers had not wanted to risk injuring the hostages by firing at their fleeing captors.

Speaking to NBC News, the DG ISPR said: "The vehicle was immobilised with sharp shooting. We destroyed their tires. The hostages remained inside the vehicle. The driver and an accomplice managed to escape to a nearby refugee camp... We moved the hostages via helicopter to Islamabad. They were then handed over to US authorities."

Officials who spoke to Dawn earlier on condition of anonymity said that one of the kidnappers had been taken into custody, one was injured and two accomplices fled.

Explore: Everything you need to know about the kidnapping and recovery of Caitlan Coleman and Joshua Boyle

Boyle described a scene in which gunshots rang out as he, his wife and their children were intercepted by security forces while being transported in the trunk of their captors' car.

Boyle told his parents there was a shootout in which some of his captors were killed and said the last words he'd heard from the kidnappers were, "kill the hostage", his father, Patrick told reporters after speaking with his son.

Late Thursday, a US military official told AFP the couple was hesitating to board a US military jet in Pakistan over Boyle's concerns he could face American scrutiny over his previous marriage to the sister of a Guantanamo detainee.

Canadian and US officials have said Boyle is not being investigated.

Pakistani and US officials confirmed the couple had left the country on condition of anonymity Friday afternoon, but gave no further details.