Through the long, busy months of autumn, the calls kept coming in: Mothers losing grip of their children while trying to cross the treacherous waters of the Rio Grande. Pregnant women getting caught in barbed wire. Bodies washing up on shore.

Cities like New York and Chicago have struggled in recent months to accommodate the busloads of migrants arriving during the latest surge in migration. But here on the border, the small town of Eagle Pass, Texas, has been one of several cities facing an even more difficult challenge. Up to 5,000 migrants a day were crossing the border there from Mexico during the height of the influx in recent weeks, gathering along the river, running through people’s yards and looking for help.

Many are in urgent need of medical attention when they arrive — help that is only available through a city that is already straining to meet the needs of its own 28,000 residents. The city has had to assign one of its five ambulances full time to transport injured migrants from the river’s edge.

“A lot of the attention is directed to the big cities and the disagreement between politicians and not the boots on the ground here in Eagle Pass,” Manuel Mello III, the Eagle Pass fire chief, said this week as a delegation of 60 Republican congressmen, including House Speaker Mike Johnson, gathered in town at the edge of the Rio Grande to call for the Biden administration to stem the immigration surge.

“Our communities are overrun,” Mr. Johnson told reporters on Wednesday. “They have opened the border wide to the entire world.”

An estimated 300,000 migrants were apprehended along the southern border in December, a record, prompting the temporary closure of four international crossings, including one in Eagle Pass.

Texas has long been at the center of U.S. immigration policy, with some of the largest levels of arrivals in recent years occurring in El Paso and across the Rio Grande Valley. But the area has turned into a key point of friction this year between Republican leaders and the Biden administration, as Gov. Greg Abbott of Texas has openly defied federal authority and set up state law enforcement patrols, concertina wire and floating buoys along the border in a bid to keep new migrants out of the state.

On Wednesday, the Justice Department filed suit challenging a new Texas law that gives state police officers authority to arrest and turn back migrants and is set to go into effect in March.

Governor Abbott is continuing to send migrants to northern cities, but local officials on the border say they are the ones being overwhelmed by the thousands of migrants arriving at their doors, especially emergency departments called to respond when there is trouble.

In Eagle Pass, most calls are for migrants who underestimate the strong currents of the river and disappear under the water. Others are for people injured trying to cross through the barbed wire that the state installed over the objections of federal authorities, pregnant women in distress, heat exhaustion, hypothermia and various minor injuries.

The Fire Department responds to an average of 217 such calls a month, a significant number for a crew of about 50 paramedics and emergency medical technicians. “We were seeing drownings almost every day,” Chief Mello said. “We are being overwhelmed.”

The department is spending about $150,000 a month on ambulance costs responding to migrants alone, a cost that is normally covered by patients or health insurers but not in the case of migrants, the chief said. That figure does not include overtime, which is costing more than $30,000 a month, and the costs of replacing equipment and medicine.

“But what’s the solution?” Chief Mello said. “Not to respond to calls? That would be inhumane. Our job is to save lives.”

The Fire Department is not the only local agency feeling the strain. The Eagle Pass police chief, Federico Garza, says his small police force of 74 officers is often diverted from everyday duties to respond to calls of migrants idling on a corner or crossing through a back yard. The sheer number of calls “can be overwhelming” he said.

Local officers are expected to turn over migrants to the U.S. Border Patrol. But those dynamics may change in March when the new state law takes effect and local officers are empowered to conduct more widespread migrant arrests, a provision that the Justice Department says unconstitutionally usurps federal authority.

Chief Garza has his own concerns about the law. He said his officers were not trained to deal with migrant encounters, and no one from the state had reached out to provide training and resources.

“Are we going to add border patrols? Those are the answers I’m waiting for,” he said. “My deal is to keep the city safe and avoid interference with them crossing and interfering with our citizens,” he added, referring to the migrants.

Leandro Gonzalez, 32, a longtime resident of Eagle Pass who lives near the Rio Grande, says he sees hundreds of people who cross into the city and wait in clusters to be picked up by Border Patrol officers near his home. They leave behind piles of debris, clothing and discarded plastic, he said. “That’s been happening for a few years now. You don’t know who they are,” he said. “You feel unsafe.”

After the record-setting numbers in recent months, there have been some signs of a decline. Apprehensions decreased to about 2,500 on Jan. 1 from 10,000 on a single day in December, according to government officials. Rolando Salinas Jr., the mayor of Eagle Pass, said his city had recently seen less than 500 in recent days.

“But that’s happened before, right? When you have several days of very little people and then all of a sudden, you have a big surge,” Mr. Salinas said. “I hope we don’t go back to seeing 4,000 or 5,000 people. That’s what creates chaos.”

The reasons for the recent slowdown are unclear, though it appears that one factor may be new steps that Mexico has taken to curb migration after the U.S. secretary of state, Antony J. Blinken, and other top American officials urged Mexico’s president, Manuel López Obrador, to intervene. Since then, Mexican officials have set up checkpoints to intercept a large caravan making its way north and have also begun deporting Venezuelan migrants, which lately have made up a large portion of new arrivals to the United States.

U.S. authorities have also stepped up deportations, with more than 460,000 migrants, including 75,000 families with children, flown to their home countries since May.

The drop-off in crossings has already lifted a weight off the town of Eagle Pass, Mr. Salinas said. The recent closure of an international crossing known locally as Bridge 1 crippled a local economy that benefits from a steady stream of Mexicans who legally cross every day to eat at restaurants, fill their gas tanks and commute to work. The closure over the holiday season cost the city about $1 million, Mr. Salinas said.

The bridge reopened at 7 a.m. on Thursday. Other crossings that had been temporarily closed were also reopening, in San Ysidro, Calif., Nogales, Ariz., and Lukeville, Ariz.

Mr. Salinas said he received a personal call this week from the Homeland Security secretary, Alejandro N. Mayorkas, about plans to reopen the bridge.

“That last thing that we want is for a bridge to be closed,” Mr. Salinas said. “That impacted local people and our pocketbooks.”

He said he also hoped the visit this week by House Republicans would put pressure on the president and Democrats to agree to stricter immigration policies.

But the emergency crew members at the Eagle Pass fire station said they did not have time to wait for the politicians in Washington to act, with lives at stake.

Harish Garcia, 27, an emergency medical technician, shared a harrowing story from the fall. During the most recent surge, he said, his crew responded to a call of a mother and daughter, about 4 or 5 years old, who were not breathing after trying to cross the Rio Grande. Mr. Garcia said he helped perform C.P.R. on both victims, loaded them into an ambulance and took them to a hospital.

He said he saw it as a win, after hearing that “they had made it.” But listening to his story, Chief Mello interjected. Both had perished later at the hospital, he informed him.

Mr. Garcia cast down his eyes. “We see it almost every day here,” he said. “It’s tough on us. But we lean on each other. It’s part of the job. We help who we can and then we have to get everything back in service and be ready for the next call.”

Miriam Jordan contributed reporting.

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