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.

Sum: Thousands of villagers have fled their homes in Indian-administered Kashmir as Indian and Pakistani troops continue to exchange fire in the region.
Q: The Malta-based, Migrant Offshore Aid Station, said it coordinated the operation along with Italian, Irish and German ships.
Italy's coastguard could not confirm the numbers, but said it was trying to help around a dozen other vessels.
The HMS Bulwark, a British Royal Navy ship, is also making its way towards Libya to assist with the situation.
BBC correspondent Jonathan Beale, who is on board, said the ship's crew would launch helicopters and landing craft on Sunday morning in the expectation they would pick up more migrants.
The Bulwark has already saved almost 1,800 people over the past month.
Italian navy ship Driade rescued 560 of the migrants on Saturday, including women and children, while the Irish vessel Le Eithne picked up 310 people.
The numbers of migrants attempting to cross the Mediterranean in the hope of reaching Europe, has increased by more than 10% in the first five months of 2015.
Italy's government predicts a total of 200,000 will arrive on its shores this year, up from 170,000 in 2014.

A: European warships and coast guard vessels have rescued more than 2,000 migrants off the coast of Libya.
Text: Latest Home Office figures showed 4,800 people were caught in 2015.
It is the third straight fall from a 2012 high of 7,900 and the first time in 14 years of records that the figure was below 5,000.
Road safety charity Brake called for the drink-drive limit to be reduced further.
Brake spokeswoman Lucy Amos said: "These statistics appear to show that people are finally getting the message that they cannot get away with drinking any amount of alcohol before getting behind their wheel.
"However, despite the fall in drivers testing positive or refusing a breath test on Wales' road there is still more work to be done.
"With traffic police numbers on the decline, it's leaving those who enforce the law with very little resources to catch those who do break the law."
Between 2002 and 2012, the average number of people failing tests was 6,520.
Insp David Cust, from North Wales Police, said he believed the fall in numbers was down to a more targeted approach.
He said: "We are using our intelligence and information from the public to look at our patrols and put them in the right areas at the right time.
"The drink drive message is getting across and the figures are encouraging, however, we can't be complacent as the drug [driving] figures are going up."
Gwent Police Ch Insp Huw Jones said he was pleased the figures were falling.
"This shows that drivers are thinking before driving under the influence. Drink and drug driving ruins lives all year round, and our efforts to combat it continue throughout the year," he added.
A Dyfed-Powys Police spokeswoman said: "We're not resting on our laurels and we will continue to enforce drink driving all year round with heightened activity during our two major campaigns in the summer and winter."
The other two Welsh police forces have been asked to comment.
For a breath test in England and Wales, the limit is 35 micrograms of alcohol in 100 millilitres of breath, while in Scotland it is 22.
Ms Amos believes the rest of the UK should follow suit.
"Drink driving is still one of the biggest killers on our roads and while we welcome these reductions, Brake is calling on the UK government to bring England and Wales into line with Scotland's lower drink-drive limit," she added.
summary: Motorists in Wales are "finally getting the message" about drink-driving as the number of people who failed or refused breath tests fell to its lowest levels.
Article: The Henblas Square site housed a post office branch, toy and clothing shops while it adjoined the BHS store that closed after the company went bust.
Manchester-based MCR Property has now acquired the large complex.
The company is working with Wrexham council officials to find tenants, with a spokesman saying it is looking at "mixed-use".
Wrexham councillor Terry Evans, lead member for economic development, called the area "the heart of the town".
"We've always said that this part of town is crucial, forming the traditional town centre alongside High Street, Charles Street, Regent Street, Hope Street and Chester Street," he added.
++++++++++
Summarize:
A major Wrexham town centre shopping centre that has been empty for a number of years is set to be reopened.