Migrants start forming a line to be picked up by the Border Patrol at a gate in the border fence after crossing from Ciudad Juarez, Mexico into El Paso, Texas, in the early hours of Thursday. Andres Leighton/Associated Press

EL PASO, Texas — The Biden administration Thursday vowed that its border strategy would succeed over time despite a record influx of migrants across the U.S.-Mexico border as it lifts pandemic-era measures and overcrowding worsens in government facilities.

Illegal border crossings have topped 10,000 per day this week, the highest levels ever, as the Title 42 border policy was set to expire at 11:59 p.m. Thursday. Thousands of migrants forded the Rio Grande into the Brownsville, Texas, area, or arrived elsewhere, including more than 800 miles away on the dusty strip of U.S. land between the riverbanks and the border wall east of downtown El Paso.

With Border Patrol stations and processing centers maxed out, officials authorized the release of migrants without court dates at locations where facilities exceeded 125 percent of their holding capacity or other thresholds were surpassed. But a federal judge stepped in late Thursday to block the release plan, granting a temporary restraining order sought by Florida’s attorney general.

The ruling cuts off a potential pressure valve for the Biden administration after senior officials have repeatedly assured they are prepared for the strains and have a broader strategy that will reduce illegal crossings.

“We are clear-eyed about the challenges we are likely to face in the days and weeks ahead, and we are ready to meet them,” Homeland Security Secretary Alejandro Mayorkas said at the White House Thursday.

“We prepared for this moment for almost two years, and our plan will deliver results,” Mayorkas said. “It will take time for those results to be fully realized. And we all must take this into account.”

For the past three years, U.S. border agents have used pandemic-related rules, launched during the Trump administration, to summarily expel border crossers to Mexico or their home countries. The Biden administration is replacing Title 42 pandemic measures with a new emergency policy that will make it easier for authorities to deport asylum seekers who cross illegally while expanding opportunities for migrants to apply to reach the United States legally.

In El Paso, thousands of migrants have been streaming toward two massive gates, two miles apart, that open into the United States. U.S. authorities have been directing migrants toward the border wall gates, marked Door 40 and Door 42, turning the river plain east of downtown El Paso into a massive outdoor waiting room.

“We just want to pass, but this process is so slow,” said Jesus Juárez, 26, of Venezuela, who said he had spent a month crossing multiple countries to reach a spot outside Door 40.

Migrants arrive on the Mexican side of the bank of the Rio Grande River in Matamoros, Mexico, on Thursday. Pandemic-related U.S. asylum restrictions, known as Title 42, expired May 11. Fernando Llano/Associated Press

He and his cousin Carlos Juárez, 31, had raspy throats and bloodshot eyes from spending hours in the swirling dust and baking sun. U.S. agents periodically opened the gates to allow groups of 10 or 15 to enter for processing and a chance to seek U.S. protection. “But 100 more arrive,” Jesus Juárez said.

The White House set the Title 42 emergency public health policy to expire Thursday, and for months U.S. officials have predicted a migration surge would occur after they were no longer able to rapidly expel border-crossers as part of pandemic restrictions.

The mere anticipation of the policy’s end triggered a race to the border this week, as tens of thousands of migrants have been crossing, with some saying they fear they’re more likely to be deported after the measure lifts.

A memo issued Wednesday by Border Patrol Chief Raul Ortiz authorized supervisors in overcrowded areas to quickly release migrants using an authority he called “Parole with Conditions.” Migrants will be vetted and directed to report to U.S. immigration authorities in their destination cities within 60 days.

According to the Ortiz memo, “the decision to parole a noncitizen must still be made on a case-by-case individualized basis, examining all the facts and circumstances at the time of the noncitizen’s inspection, and only if there is an urgent humanitarian reason, such as ensuring the safety, health, and security of the individual noncitizen, or significant public benefit justifying parole.”

The Border Patrol has averaged 8,750 migrant encounters per day over the past week, the memo noted, more than twice as many as the peak of the 2019 crisis when record numbers of Central American families overwhelmed the Trump administration.

Migrants wait in the cold at a gate in the border fence after crossing from Ciudad Juarez, Mexico, into El Paso, Texas, in the early hours of Thursday. Andres Leighton/Associated Press

But U.S. District Judge T. Kent Wetherell ruled late Thursday that the new release plan was “materially indistinguishable” from a similar policy he deemed unlawful in a March decision siding with the state of Florida.

“In both instances, aliens are being released into the country on an expedited basis without being placed in removal proceedings and with little to no vetting and no monitoring,” Wetherell wrote. His previous decision said the Biden administration had turned the border “into little more than a speed bump” for migrants arriving illegally.

Wetherell’s order will remain in effect for 14 days.

Customs and Border Protection officials had more than 25,000 migrants in holding cells and processing facilities along the border this week, three times the system’s capacity, according to the latest government data obtained by The Washington Post.

Thousands of mostly-Venezuelan migrants have been crossing the Rio Grande in the Brownsville area, many using inflatable flotation devices. Texas National Guard troops and state police deployed by Gov. Greg Abbott, R, have at times tried to block migrants from climbing up the river banks, but many of the migrants in the Brownsville area appeared to be getting through to line up for processing at Border Patrol tents.

At the opposite end of the state, in El Paso, where the Rio Grande is little more than an oozing trickle, a well-worn path from the Mexican side leads down an embankment to hopscotch across river stones and a junked wooden pallet.

Once migrants cross the center of the river – away from an official port of entry – they have entered the United States from Mexico illegally. They walk up the U.S. side through an opening in the razor wire, joining the mass of people who have been waiting in international limbo, at times crossing back into Mexico for supplies.

Migrants have complained they see no international agencies coming to their aid as they wait in the roughshod encampments they’ve set up in this peculiar corridor of land divided from the United States by the bars of the border wall.

“No one has helped us. No one at all,” said Angel Moran, a 50-year-old Venezuelan who said he’d been waiting six days. “I have never seen an international group here. And there are sick children breathing dust, all day and all night. I feel helpless watching that and not being able to do anything at all to help. Who is supposed to help these children and their mothers?”

Moran said he was the coordinator of the Mesa de Unidad Democrática organization in Venezuela – a group opposed to Venezuelan President Nicolás Maduro and his predecessor Hugo Chávez – so he considered his need to flee urgently.

At shelters on the Mexican side and along the streets of Ciudad Juárez, migrants faced a choice: cross now, or wait for a tense situation to pass.

Severino Ismael Martinez Santiago, the director of the Pan de Vida shelter in Ciudad Juárez, the Mexican border city opposite El Paso, said many migrants were confused about what the lifting of Title 42 rules would mean for their effort to gain legal entry into the United States.

“They are misinformed. They believe that when Title 42 ends, the doors to the United States will be opened, and they can cross,” Martinez said. “But that is far from the truth. It will be far worse for them.”

Biden administration officials say they will ramp up deportations and impose new restrictions on asylum seekers who cross illegally when Title 42 lifts.

Martinez said people considering migrating to the border of the United States should wait, if possible.

“See what happens,” he said. “Right now, things are up in the air with how to enter the United States. Why risk your life, the lives of your children? Why come, if suffering awaits you? Wait six months to see how things are going.”