Civilians fleeing the fighting between two rival generals in Sudan streamed into neighboring countries on Monday, raising concerns about a humanitarian crisis spreading to countries already grappling with conflict, hunger and dire economic straits.

The heavy gunfire, shelling and airstrikes that have rocked Sudan for 10 days prompted foreign countries to begin evacuating diplomatic staff and nationals over the weekend. It also has driven thousands of Sudanese and other people across borders into Chad, Egypt and South Sudan, aid workers said.

The huge movement of people risks overwhelming Sudan’s neighbors, some of which already host large numbers of refugees and internally displaced people. Sudan, a country of 45 million people and the third-largest by area in Africa, is surrounded by seven countries racked by poverty and instability.

By Monday, nearly 3,000 people had arrived in the South Sudanese border town of Renk, according to the International Organization for Migration, a United Nations agency. Most of those were South Sudanese returning home after having fled Khartoum in cars and on the backs of trucks, carrying whatever they could on the 280-mile journey south.

“The people that get out first are the people that have the means,” Peter Van der Auweraert, the South Sudan representative for the International Organization for Migration, noted. “We are preparing for more vulnerable people arriving in the coming days and weeks.”

As the fighting in Sudan rages, more than 400 people have been killed and 3,700 others injured, according to the World Health Organization. The clashes have left countless people in the country without food, water or electricity. Many hospitals in the capital, Khartoum, have closed and several humanitarian organizations said that their warehouses and offices had been looted.

Repeated efforts to broker a cease-fire between the two rival forces — the army and the Rapid Support Forces, a paramilitary unit — have failed.

Hundreds of families, according to relatives and aid workers, are fleeing to small cities and towns in the eastern and southern parts of Sudan, and some are considering crossing into Ethiopia — which is still recovering from two years of civil war in the northern Tigray region that was quieted only last November.

In Sudan, foreign countries began to evacuate diplomatic staff over the weekend in airlifts and long convoys by car to Egypt or a port on the Red Sea. But they have left behind a pool of resentment on the part of some Sudanese, who say they feel both abandoned and angry that international diplomacy failed to prevent the military rivals from turning to battle.

The removal of foreign nationals continued on Monday, with the European Union evacuating 21 diplomatic staff and “more than 1,000 civilians,” according to the bloc’s top diplomat. Several African nations, including Djibouti, Kenya, Nigeria and South Africa also announced plans to evacuate their nationals.

The United Nations mission in Sudan said that its international staff had reached the city of Port Sudan on the Red Sea and would go from there to neighboring countries. However, Volker Perthes, the U.N. envoy to Sudan, will remain in the country.

Even as some found a way to leave the most dangerous areas of the country, many people remain stuck in Khartoum, where the conflict is most acute. Javid Abdelmoneim’s father, an 80-year-old British national who lives in the Garden City district of Khartoum, was among them.

Mr. Abdelmoneim said that his father had declined offers from relatives to leave the city because he said that he had received promises from the British Embassy that he would be evacuated. But the British government evacuated only its diplomatic staff on Sunday, a move that Mr. Abdelmoneim said he and his family had learned about on Twitter.

In a phone interview from the Malawian city of Blantyre, where he works, Mr. Abdelmoneim said that he had been unable to reach his father on Monday. He said that he was coordinating with relatives abroad to reach two uncles who were departing Khartoum and might be able to take his father with them but that phone and internet networks were patchy.

“It is a precarious situation,” Mr. Abdelmoneim said, through tears. “My only hope is that we will somehow reach our relatives and have them get my father out.”

Complicating matters further, Sudan has been hosting about 1.1 million refugees and asylum seekers, according to the United Nations refugee agency. Most of those people are from South Sudan, a nation that split from Sudan in 2011 and has been ravaged by civil war ever since. Sudan is also home to refugees from conflicts and autocratic rule in countries such as Eritrea, Ethiopia and Syria.

The violence has now forced some of those refugees and asylum seekers to take to the road again and cross into neighboring countries. For many of those people, who had just begun to pick up the pieces of their lives by opening small businesses and otherwise putting down tentative roots in Sudan, any hope of regaining a settled existence is now in limbo again.

The countries they are fleeing to are themselves vulnerable. In just the past few years, there has been the civil war in Ethiopia, hunger and economic challenges in South Sudan, and a coup in Chad, for instance. Aid workers have warned that a broader displacement of people fleeing the fighting in Sudan could have a disastrous effect on those neighboring nations.

When the fighting broke out in Khartoum, pockets of violence also flared up in the western Darfur region. That sent up to 20,000 people — mostly women and children — fleeing into neighboring Chad, which is already home to more than 400,000 Sudanese refugees, according to the U.N. refugee agency.

South Sudan, one of the poorest nations in the world, is already bracing for what could be a catastrophic economic shock. While most of the South Sudanese living in Sudan are refugees, the rest are migrants who typically support their families back home. The fighting could interrupt those flows of money and limit cross-border trade.

Markets in the north of South Sudan, filled with goods brought in from Sudan, already have less to offer as fighting disrupts the supply chain, Mr. Van der Auweraert of the International Organization for Migration said. And the South Sudanese pound has begun to lose value.

South Sudan, the world’s youngest nation, is wrestling with its own internal problems, particularly stemming from the yearslong civil war that has devastated the economy, cost the lives of more than 400,000 people and displaced 4.3 million others. About three-quarters of the population, or more than nine million people, are in need of humanitarian aid, according to the International Organization for Migration.

“We do not want to get into a situation where we have to deprive people in South Sudan that are also in need,” said Mr. Van der Auweraert, with the International Organization for Migration. “There’s going to be difficult decisions to be made.”

Matina Stevis-Gridneff contributed reporting.