At least 10 Indian and Pakistani soldiers and civilians have been killed in the violence over the past week.
Both sides have accused each other of starting the hostilities.
A ceasefire agreed in 2003 remains in place, but the neighbours often accuse each other of violating it.
The latest round of hostilities come days before US Secretary of State John Kerry is due to visit India. President Barack Obama is also due to visit India as the chief guest at the country's Republic Day celebrations on 26 January.
Some 10,000 civilians living in border villages on the Indian side have fled their homes since fighting began last week, a senior official named Shantmanu told Reuters news agency.
"We had a narrow escape and there is a war-like situation," Sham Kumar, 54, an affected villager said.
"Pakistani troops are using long-range weapons. It is the first time we have seen such intense shelling."
He said he had left his village after a shell landed in a school about 3.5km (2 miles) from the border.
India and Pakistan are continuing to exchange fire in the disputed region on Tuesday, reports say.
"The firing is going on and we are giving befitting reply to Pakistani shelling," an Indian official said.
Pakistan said on Monday that its soldiers were "effectively responding to India's [unprovoked] firing".
A Pakistani military statement said four civilians, including a woman and a teenage boy, had been killed by Indian fire near the city of Sialkot since Sunday.
India said one of its soldiers was killed by Pakistani fire in the Samba area on Monday.
Last week, Indian forces in Kashmir killed four Pakistani troops on the border after an Indian soldier was killed in an attack blamed on Pakistan. Pakistan said it had lodged a protest with the Indian high commissioner in Islamabad.
Hostilities between the neighbours have escalated in the past year.
In October, 16 people - nine Pakistanis and seven Indians - died when the two sides exchanged fire for several days.
Correspondents say 2014 saw an escalation in hostilities between the neighbours, with some of the worst violence in a decade. Both sides have accused each other of initiating the clashes.
Earlier in the summer, India's new Prime Minister Narendra Modi invited his Pakistani counterpart Nawaz Sharif to his inauguration and there were hopes that relations between Delhi and Islamabad would improve.
But relations have deteriorated since then, with India cancelling scheduled talks with Pakistan in August and insisting that Delhi would "not tolerate acts of border violations by Pakistan" and that "ceasefire violations must stop".
Kashmir, claimed by both countries in its entirety, has been a flashpoint for more than 60 years and the South Asian rivals have fought two wars over the region.
Thousands of villagers have fled their homes in Indian-administered Kashmir as Indian and Pakistani troops continue to exchange fire in the region.