McALLEN, Tex. — Exhausted and dazed, hundreds of Central American migrants, mainly women with small children, come to the bus station of this border city every day now, spilling into a church next door that has opened its doors. Having crossed into the United States illegally, the new arrivals are often grimy and famished. In the church, they eat, bathe and sleep, changing into donated shoes and clothes.

With no immigration detention site equipped for women with children in the area — the closest one, in Pennsylvania, is overbooked — they are freed by the Border Patrol with a bus ticket to travel to where they have relatives in this country, and an order to appear in immigration court in 30 days.

They are among at least 30,000 migrants released this year, border officials and federal lawmakers said, amid a surge of illegal crossings in the Rio Grande Valley.

While most men are held and processed quickly for deportation, border authorities struggling to manage the influx have been releasing pregnant women and parents with young children, allowing them to join family members living here and issuing them a deportation hearing notice. Migrants have sent word back home they received a “permit” to remain at least temporarily in the United States, feeding rumors along migrant routes and spurring others to embark on the long journey.

“I heard in Guatemala that people were caught by immigration, but then they let them go and gave them a permit,” said Carmen Ávila, 26, who is seven months pregnant and came with her 4-year-old son, Jostyn. “The word got around and that’s why so many people are coming.”

Migrants here said they planned to attend their court hearings and fight for a chance to stay. But officials have no specific plan to monitor compliance, and based on the pace of the overburdened immigration courts, it seems highly unlikely that any of the migrants would be deported soon.

The Obama administration has focused on the sudden increase in South Texas of young migrants traveling without their parents, calling it a humanitarian crisis and naming the Federal Emergency Management Agency to coordinate shelter and care. But the unaccompanied youths are part of a larger flow of Central Americans that officials say also includes unprecedented numbers of families with small children.

On Saturday, Representative Henry Cuellar, a Democrat who represents a South Texas district, said he saw nearly 1,000 migrants of all ages packed when he visited the Border Patrol station (normal capacity: 400) in Hidalgo.

After the Border Patrol began to release migrants in McAllen early last week, hundreds came to the refuge that community volunteers and city officials hastily set up in the Sacred Heart Catholic Church.

In Washington, the increase in illegal immigration has provoked a new argument between the White House and Republicans. Obama administration officials insist that factors in Central America, including poverty and criminal violence, are driving the migrants. Republicans blame lax enforcement by the administration. Representative Robert W. Goodlatte, Republican of Virginia and chairman of the House Judiciary Committee, will hold a hearing next week on what he described as “an administration-made disaster.”

At the church, some women said the talk about an entry permit, which has intensified in the last two months, had prompted them to set out on the risk-filled journey across Mexico. But the women said they were moved mainly by desperate worries about their children, with poverty unrelenting in their countries and warring street gangs expanding their control.

The Central American migration has created unusual difficulties in the Rio Grande Valley for border authorities, who must follow differing rules for unaccompanied minors; for migrants who are not from Mexico; and for women who have children or are pregnant.

Since October more than 47,000 unaccompanied youths have been apprehended along the Southwest border, and border officials estimate that number may double by the end of this year. The Border Patrol is required to transfer unaccompanied youths within 72 hours to a refugee agency in the Department of Health and Human Services, which runs shelters for them and works to locate parents or guardians in the United States. Federal emergency officials have opened new shelters for the children at three military bases and are coordinating food, medical care and legal assistance.

From the shelters, youths are sent to live with parents or relatives here or to longer-term foster care if they have no family here and have grounds to fight deportation. Otherwise, they face deportation.

For women with children, the Obama administration has a longstanding policy of seeking separate, family-friendly detention centers or releasing them, often without bond.

Homeland Security Secretary Jeh Johnson has emphasized that these recent migrants have no immigration status and are in deportation proceedings. If they do not win their cases, as recent border crossers, Mr. Johnson said, they would be among the first to be deported “regardless of age.”

But backlogs are huge in immigration courts, with waits of over a year for a hearing. Many women and children who made it to the United States in the past year are still here. “There’s a perception,” Mr. Cuellar said, “that if you step on American soil, they will give you a piece of paper, you go to the bus station and you can go anywhere in the United States. You’re free with that permiso,” he said, using the Spanish word for permit. Mr. Cuellar wants more resources and immigration judges to process cases in South Texas.

Smugglers have stoked the permit rumors, migrants and border agents said, since they profit from the traffic. At Anzalduas Park, a pleasant fishing spot on the river near here, smugglers run migrants across the water on motorized water scooters.

In a perplexing problem for the Border Patrol, many women and youths who cross the Rio Grande illegally now run toward agents rather than away from them, believing that being caught is the first step toward an entry permit.

In the community center at the church, children, at first hungry and timid after a grueling trip, began to scamper and play with the toys they had been given. McAllen residents have rushed to help the migrants, piling tables high with donations while city officials have supplied portable showers and bathrooms.

Meybell Ramos, 38, said she left a decent job as a social worker in El Salvador to get her 11-year-old daughter, Katherine, away from the gangs in her neighborhood.

“The gangs came to my house,” Katherine said. “They told my mother: ‘Take care of your daughter. Her body is becoming so pretty.’ ”

Her mother added: “If you don’t do what they say, those boys, they will kill you. I was ready to leave everything behind to protect the life of my child.”

Leiby Mejía, 27, who came from Honduras with two sons, 5 and 7, said she heard the permit rumor, then fled after a narcotics gang killed a cousin living nearby.

“I feel happy and sad at the same time,” Ms. Mejía said. “Happy because I’m here. Sad because I see the permit is not what I expected.” She added, “I don’t know if they will let me stay or not.” “

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