PESHAWAR, Pakistan — Pakistani helicopters and artillery today forced back militants who crossed over from Afghanistan this week and triggered battles that have killed scores of people, an official said.

The government demanded that NATO and Afghan troops do more to control insurgents on their side of the long, porous border.

The Pakistani Taliban, the country’s most prominent militant group, claimed responsibility for the incursion in a telephone call to reporters. If true, the group’s presence in Afghanistan’s Kunar province marks a new demonstration of the instability of the border area 10 years after the U.S.-led invasion of Afghanistan.

Taliban spokesman Ahsanullah Ahsan said between 40 and 50 fighters took part in the raid, much less than the 400 claimed by the government. He said just two of the militants died and that many Pakistani troops were killed.

The insurgents entered Pakistan’s Upper Dir region from Kunar province on Wednesday. They attacked a security checkpoint, villages and schools, according to the Pakistani government.

Regional administrator Ghulam Mohammad Khan said the militants were retreating Friday and under Pakistani attack in one district.

As of Thursday night, 25 soldiers, 35 militants and three civilians had died in the clashes, Khan said. He had no information about casualties from Friday’s fighting. Reporters were blocked from traveling to the scene of the fighting.