Tension between the nuclear-armed rivals has risen since Indian Prime Minister Narendra Modi called off peace talks in August and clashes along stretches of their border have been erupting intermittently since October.
At least 10 Indian and Pakistani soldiers and civilians have been killed in fighting over the past week.
About 6,000 civilians in Indian-controlled Kashmir fled from their homes late on Monday as fighting moved to civilian areas, said Shantmanu, the divisional commissioner of Jammu region. About 4,000 left after fighting began last week.
It is alleged that Pakistani troops are using long-range weapons in what is believed to be the first time such shelling has taken place. The violence comes days before U.S. Secretary of State John Kerry is due to visit India. President Barack Obama is also due to visit India later this month.
The United States has for years been trying to push the South Asian rivals to build better relations. Mistrust between India and Pakistan is a factor behind conflict in various parts of the region including Afghanistan.
Kumar said he left his village after a shell landed in a school about 3.5 km (2 miles) from the border.
Indian and Pakistani forces again exchanged gunfire and mortar bombs across parts of their border on Tuesday, an Indian Border Security Force official said.
“The firing is going on and we are giving befitting reply to Pakistani shelling,” the official said.
Pakistani officials were not immediately available for comment.
The rivals, who have fought two wars over Muslim-majority Kashmir, blame each other for the upsurge in clashes since October.