Tens of thousands of Venezuelans are stranded in Mexico and Central America after U.S. officials applied a Trump-era policy to deter a deluge in illegal border crossings by Venezuelan migrants that for months had vexed the Biden administration.

The United Nations estimates that the U.S. has since Oct. 12 expelled more than 5,300 Venezuelans who had arrived at the border back to Mexico under Title 42, first implemented under President Donald Trump to permit the expulsion of migrants on grounds they might be positive for Covid-19. Prior to that, Venezuelans had been leaving Venezuela or third countries in record numbers to reach the U.S.

The Venezuelans, many with children, are now sleeping on the streets, in makeshift camps and at overcrowded shelters on the Mexican side of the border, challenging the country’s federal government and local authorities, Mexican officials said. The migrants were expecting to cross and plead asylum, then remain in the U.S., like tens of thousands of other Venezuelans who migrated earlier this year .

Now, many of the Venezuelans say they are despondent, hungry and out of money after traversing several countries and dodging violence and thieves only to get shut out of the U.S.

“We had tremendous confidence, we sold everything to get here and suddenly the door slammed in our faces,” said Félix Rodríguez, a Venezuelan horse trainer who headed north in late September from Argentina, where he had first fled. He spoke from a rundown motel in Piedras Negras, a dusty border community across from Eagle Pass, Texas.

In the campaign in the U.S. ahead of next month’s midterm elections , Republicans made President Biden’s handling of immigration a key line of attack . Now, the Biden administration is highlighting the results of its new policy. U.S. immigration officials said illegal border crossings by Venezuelans fell from about 1,200 a day earlier this month to 150 a day in recent days.

But the administration’s shift in policy has angered immigration and human-rights advocates, and even raised questions among some Democrats, who say it expands a Trump-era policy the administration had aimed to dismantle while withdrawing the possibility for many Venezuelans to apply for asylum at the border.

“We cannot as a nation throw away what is the essence of asylum, because then you cannot be a beacon of light to other countries,” Sen. Ben Cardin (D., Md.), a member of the Senate Foreign Relations Committee, said on a recent trip to Latin America.

But Homeland Security Secretary Alejandro Mayorkas has said that the measures aim to provide a lawful and orderly way for Venezuelans to enter the U.S.

The change came after apprehensions of migrants at the U.S. southern border hit a record 2.2 million in the fiscal year that ended Sept. 30. They included more than 187,000 Venezuelans, a fourfold increase over the previous year, according to U.S. Customs and Border Protection data. In August and September, Venezuelans made up the second-largest nationality to cross the southwestern border illegally after Mexicans.

Venezuela’s economic crisis—marked by galloping inflation, poverty and criminal violence—has led 7.1 million Venezuelans, a quarter of the population, to flee in recent years, the U.N. recently said. Most migrated to neighboring countries , but hardship has driven thousands north to the U.S. border as word spread that Venezuelans migrants could surrender to U.S. authorities and request asylum without facing expulsion, since Venezuela’s government doesn’t accept deportees.

In recent days, large groups of Venezuelans thwarted from reaching the U.S. have crowded shelters in Mexican border cities like Ciudad Juárez. They have arrived in large numbers in the small community of San Pedro Tapanatepec in southern Mexico, which is used by Mexican officials to hold migrants entering from Guatemala. There are also thousands stranded as far south as Panama. Many were on their way to the U.S., having crossed the dangerous jungle known as the Darien Gap between Colombia and Panama, when word came about the policy shift.

“When the news reached us via WhatsApp and Facebook, we suddenly began to see dozens of people crying in the street,” said Kevin Camejo, a former police officer in Venezuela who was stuck in Piedras Negras.

The only legal option now open to Venezuelans is the administration’s “Volunteering for Venezuela” program, allowing up to 24,000 Venezuelans to apply online for admission if they have a U.S. financial sponsor and pass national security checks. Venezuelans who qualify must enter the U.S. at airports and won’t be allowed to use ports of entry at the Mexico-U.S. border.

More than 7,500 Venezuelans have applied since the program went online last week, and about 150 had been approved to travel, according to Mexican and U.S. officials. In recent days, a handful of Venezuelans have arrived at U.S. airports.

But most migrants likely won’t qualify for the program, which requires them to hold unexpired Venezuelan passports and not to have crossed a border illegally since early October. For most Venezuelans, accessing a passport is difficult, expensive and can require an indefinite waiting period for processing.

Jonathan Hooka, a former bodyguard from one of Caracas’s most-dangerous barrios, said he fled Venezuela on foot two months ago with his wife, trekking through Colombia and the Darien Gap, then crossed Central America before joining a caravan of about 400 migrants that reached the U.S. border.

After surrendering to U.S. authorities, Mr. Hooka said he was shuttled between seven different detention centers over five days before being expelled to Mexico. Mr. Hooka said he and his wife, now at a shelter in Mexico City, would try applying for a U.S. visa through the new program, although they have no idea how long it will take.

“Going back to Venezuela isn’t an option,” Mr. Hooka said.

Local government officials along the U.S.-Mexico border and several Venezuelan migrants interviewed in Mexico and the U.S. doubt that the new measure will deter migration. They note that the number of visas that would be handed out is well below the almost 34,000 Venezuelans apprehended on the U.S. southern border in September alone.

Sergio Valderrama, a Venezuelan migrant who crossed the U.S.-Mexico border in June and works at a vegetable processing plant in Virginia, said his brother had been among Venezuelans who had crossed the Darien Gap this month when the new policy was announced.

“The migrant route is already open, and it’s difficult to go back,” Mr. Valderrama said. “Many will likely find a way to cross the border, like Mexicans or migrants of other nationalities do.”

Migrants expelled under Title 42 face no penalty for repeated attempts at crossing illegally, which lead many to attempt multiple times, according to U.S. government data. “Title 42 has created a revolving door,” said Sen. Robert Menendez, (D., N.J.), chairman of the Senate Foreign Relations Committee. “They can come to the border. Two times. Ten times. A hundred times.”

Smugglers are also likely to take advantage of the new policy, since migrants will search them out to help them cross the border.

Luis Conde, a Venezuelan, was stuck in the Mexican capital, looking for a new plan after being deported from the U.S. earlier this month. He said he spent much of his life savings making the nearly 4,000-mile trek from Peru. Now, smugglers at a shelter were offering to take him across the border for $700, more than he could afford.

“This situation is just going to empower those people who want to take advantage of migrants who are trying to realize their American dream,” Mr. Conde said.