ZAATARI CAMP, Jordan — Some residents even tossed stones at the military field hospital said to be where the children died, although officials say all four succumbed to medical conditions that had nothing to do with the weather.

“These were incorrect rumors,” said Ahmad Maaytah, a doctor from the hospital, as his colleagues nodded in agreement at a camp clinic.

Still, the incident is indicative of the extraordinary pressure that the winter chill is placing on already thinly stretched international efforts to cope with an influx of more than 700,000 refugees, mostly women and children, into Jordan and other nations bordering Syria. Thousands more arrive each day, many with little more than the clothes on their backs.

They are a social and economic strain on Syria’s neighbors, and they are the most dramatic illustration of how Syria’s 20-month-old war is spilling across its borders.

Winter’s cold is a serious threat, especially for children and those already weakened after their escape from Syria, say officials with the United Nations and other agencies assisting the Syrian multitudes.

Many now arrive drenched and shivering in the rainy evenings, their shoes caked in mud. Inevitably, a number suffer from medical ailments and psychological trauma.

On a recent evening, 22 newborns and an 85-year-old woman accompanied by her grandchildren were among those crossing into Jordan, U.N. officials said.

Two of the recent infant deaths at Zaatari resulted from severe diarrhea, possibly because of poor hygiene, the U.N. refugee agency said. The other two suffered from congenital defects, one of the heart and the other of the digestive tract, according to the U.N.

At Zaatari, home to about 40,000 refugees, workers are hastening to winterize the vast camp, situated in an inhospitable stretch of desert where winter temperatures regularly plunge below freezing and strong winds and snow are not uncommon.

Officials are distributing tens of thousands of “high thermal blankets,” and a Norwegian relief group is providing portable gas heaters. Saudi Arabia and other donors have provided prefabricated fiberglass homes to allow some people to escape the canvas tents where most refugees still live.

Contrary to popular perception, fewer than one quarter of Syrian refugees live in the region’s 20 camps — three each in Jordan and Iraq and 14 in Turkey.

The vast majority of refugees stay with host families, rent homes or otherwise scramble to find housing and aid. More than 500,000 have signed up for U.N. assistance.

The crisis is especially acute for Syrians who have fled to Lebanon, where there are no refugee camps and humanitarian groups are stretched to find host families, public facilities, vacant buildings and other places to house the dispossessed. Refugees are scattered across hundreds of towns and villages — some in isolated corners of the Bekaa Valley and other remote zones, often crammed into substandard and overpriced accommodation.

“There were protests and gunfire, my children were scared, so we had to leave” Syria, said a mother of three huddled in a converted storage space in the Bekaa town of Zahle where she pays $150 a month, almost three times her former rent in the strife-torn Damascus suburb of Hajar Aswad, which she left three months earlier.

“This is where we shower and do laundry,” said the woman, 30, pointing to a bucket on a concrete slab.

But she may be better off than many other displaced Syrians. An estimated 2 million people who have been forced from their homes but remain in Syria probably suffer the worst deprivation. Some are making do by roughing it in fields, bombed-out buildings and abandoned schools. Many still face danger from bombardment and firefights as they try to cope with a lack of shelter, food, medical care and other necessities.

At Zaatari, as at other camps, life is spartan, and it is not difficult to find people complaining about the conditions.

The other day, a mother scoffed at the quality of the blankets covering her two boys, who were sleeping at noon because it is too cold to doze off in the evening.