While some residents complain of an increase in panhandlers, crime has not worsened and those who come do not generally stay, said the city manager, George Antuna. “Most of the folks that come here are going north — D.C., Chicago, New York, Miami,” he said.

But it is the sheer number lately that has been overwhelming. “We’re not equipped for this,” Mr. Antuna said.

It has even strained smuggling networks, said Mr. McCraw, the state police director. “They’re running out of drivers,” he said, pointing to interviews with those charged with smuggling and Spanish-language TikTok videos seeking drivers to shuttle migrants from the border to cities like Houston.

“My view is what we’re succeeding at is securing zones” along the border, particularly in the Rio Grande Valley, where the number of illegal crossings has declined, Mr. McCraw said. “It’s like hot spot policing.”

In Eagle Pass, Border Patrol buses with asylum seekers now arrive in a constant stream at the main respite center, which had to move from its small downtown space to a cavernous, warehouse-like building. The number of migrants seeking services, said Valeria Wheeler, the director of the center, Mission: Border Hope, has ballooned to as many as 500 a day, from about 20 a day two years ago.

Still, many migrants who arrived there this week walked away frustrated at the lack of space. Some said they had to sleep on the concrete floor.

“Where are we all going to fit?” Diego Carmona, 28, wondered after he arrived at the end of a grueling five-month journey from Venezuela with his wife, 8-year-old son and 7-month-old baby.

Mr. Carmona said that as he crossed the river he feared his older son might be swept away. He said he could still hear him screaming, in Spanish, “Daddy, I don’t want to die,” as they traversed the unpredictable current. “It was the worst moment of my life,” he said — but they made it.

It was at a bend in the river, north of downtown, that a National Guard soldier from outside Dallas, Specialist Bishop Evans, had been stationed with a partner when he spotted a man and a woman struggling in the river as they crossed from Mexico. He rushed to help them, jumping several feet off the high banks into the fast-moving water.

Specialist Evans drowned. The two migrants, whom state officials have said were involved in drug trafficking, survived and were taken into Border Patrol custody.

No National Guard members were posted on the high flat ground on a recent overcast day. Below, the river churned near a path littered with discarded clothes and other items from migrants who had recently passed through.

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