MATAMOROS, Mexico (AP) — Waiting on the southern end of a bridge that leads to the United States, Humberto Alvarez Gonzalez warily follows the progress of the caravan winding through Mexico with the goal of reaching the border.

Alvarez and about two dozen other people are waiting in Matamoros, across the Rio Grande from Brownsville, Texas, because U.S. customs officers say there's no space to process them. They sleep on cots near the bridge and rely on donors who bring them food and clothing. Some have waited for two weeks.

Now, Alvarez, a 32-year-old from Cuba, is worried that large waves of migrants in a caravan still more than 800 miles away from the border might provoke the U.S. government to reject them altogether.

"Our idea is to enter before the caravan," he said. "We are afraid that the group of migrants will reach us and that they will judge us together with them."

Asylum seekers already camping at border crossings worry that how the Trump administration responds to the caravan of some 4,000 Central American migrants and three much smaller ones hundreds of miles behind it could leave them shut out. President Donald Trump last week threatened to detain asylum seekers in large tents and send as many as 15,000 active-duty soldiers to the border. He's also spoken of closing the border.

U.S. government officials say the bridges remain open to asylum seekers. But in South Texas, the busiest corridor for unauthorized border crossings, U.S. Customs and Border Protection officers stand at the center of bridges to check documents and stop most asylum seekers.

And in San Diego, people at the San Ysidro crossing wait more than a month, and volunteers operate an informal take-a-number system to spare migrants from having to wait in line or sleep out in the open. Inspectors there typically process about 100 claims a day.

"It's not turning people away, it's asking them to wait," CBP Commissioner Kevin McAleenan said recently. "We are taking people in as we have capacity to do so."

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At the bridge where Alvarez and dozens of others wait, security guards on the Mexican side hold back asylum seekers until U.S. border inspectors tell them how many people they will accept. Some days, five or 10 people are allowed. On other days, the asylum seekers said, no one is.

At another bridge separating Brownsville and Matamoros, four women and their children slept under a blue tarp on the bridge. The tarp had been left behind by previous asylum seekers. One of the women said she had waited there for two days.

Seeking asylum at a port of entry is legal under U.S. law, as government officials have reaffirmed this year. Trump has proposed banning people crossing illegally between ports of entry from claiming asylum — something many immigration experts say he can't do under the law.

The U.S. fielded nearly 332,000 asylum claims in 2017, nearly double that of two years earlier and the highest of any country in the world, according to the United Nations High Commissioner for Refugees. And many asylum seekers wait for years to have their claims adjudicated, a delay criticized by administration officials and immigration lawyers alike.

Attorney General Jeff Sessions ruled in June that fleeing gang or domestic violence would generally not be considered grounds for asylum, a decision that would affect many people's claims.

Jodi Goodwin, an immigration attorney in Harlingen, Texas, said she believes the government was trying to frustrate asylum seekers into giving up.

"I think they're using a lot of ruse and double-speak to try to circumvent the law," she said.

Two women and three children were waiting Saturday at the center of the bridge between McAllen, Texas, and Reynosa, Mexico, a day after soldiers had installed concertina wire on the bridge and the riverbank below.

After some time, an official from Mexico's migration bureau came up to the group of five and directed them off the bridge, away from U.S. customs.

Associated Press journalists followed as the official and then a private security guard led the group back into Mexico and into a migration bureau office. An employee inside the building would not let the AP interview the group, and U.S. border inspectors on the bridge declined to comment.

At the bridge in Matamoros, Mexico, Alvarez continued to wait with others from all over the world: Honduras, Peru, Venezuela and Cameroon.

They insisted they will only enter peacefully and lawfully, waiting their turn at the bridge.

Javier Mederos Mendez, a 44-year-old from Havana, said he was fleeing political repression because he disagreed with the ruling Castro family. He said he had flown from Cuba to South America, then crossed through jungles and violence-torn areas of Central America and Mexico to reach the bridge.

He and others had seen people drown or starve along the way.

Asked what he thought of Trump's announcements about asylum, Mederos said, "It would be unfair of me to decide what is good or bad for a country."

But, he said, "I will wait as long as necessary until they receive me."

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Associated Press writer Elliot Spagat in San Diego contributed to this report.