Despite President Donald Trump's vow that a migrant caravan "better be stopped" before it reached the US, at least 88 migrants have already set foot on US soil.

The migrants are part of a roughly 200-strong group that traveled through Mexico in April, intending to seek asylum at the US border.

The Trump administration has suggested many of the migrants' claims of "credible fear" were fraudulent.

Attorney General Jeff Sessions also announced Wednesday he was sending dozens of prosecutors and immigration judges to the border to help process the backlog of cases.

Dozens of migrants in the "caravan" that traveled through Mexico in April have already set foot on American soil and applied for asylum, despite President Donald Trump's repeated demands that they be barred from entering the country at all.

At least 88 migrants have been permitted to enter this week, Reuters reported. US Customs and Border Protection held many of them off for days after they arrived at the San Ysidro port of entry in Tijuana, Mexico, saying that the border crossing had reached capacity. But on Monday, the agency began allowing some to apply for asylum.

The initial group of migrants deemed a priority were a small group of four children, three of their mothers, and one 18-year-old man, according to The New York Times. On Wednesday, CBP officials also allowed in three other groups of 63 migrants.

The migrants are part of a group of roughly 200 who also intend to seek asylum, many of whom remain in a makeshift tent encampment at the US-Mexico border across from San Diego, BuzzFeed News reported.

One organizer told CNN on Tuesday that the caravan will stay there until "every last one is admitted into the United States."

The caravan is an annual procession organized by the activist group Pueblo Sin Frontera — People without Borders. The roughly 200 migrants currently waiting at the US-Mexico border are all that remains of the original 1,000-member caravan that set out from Tapachula, Mexico, on March 25.

On Wednesday, the Justice Department announced it would send dozens of prosecutors and immigration judges to the border, in part because of the caravan.

"We are sending a message worldwide: Don't come illegally. Make your claim to enter America in the lawful way and wait your turn," Sessions told reporters. "People are not going to caravan or otherwise stampede our border. We need legality and integrity in the system."

'An attack on the sovereignty of this nation'

People who traveled with the annual caravan of Central American migrants, rest where the group set up camp to wait for access to request asylum in the US, outside the El Chaparral port of entry building at the US-Mexico border in Tijuana, Mexico, Monday, April 30, 2018. Associated Press/Hans-Maximo Musielik Trump has been railing against the caravan since it began attracting media attention in April, and raged on Monday against immigration laws that allow them to seek asylum in the US, and live in the country as their cases wind through the court system.

"The migrant 'caravan' that is openly defying our border shows how weak & ineffective U.S. immigration laws are," Trump tweeted Monday.

The Trump administration has also suggested that many of the migrants' claims are fraudulent. The acting director of the Immigration and Customs Enforcement agency, Thomas Homan, told "Fox & Friends" on Tuesday that the caravan represented an "attack on the sovereignty of this nation."

"Look, do I think some of these people have a credible fear case? Do some of these folks — are they escaping fear and persecution? Yes, some are. But I also know that some aren't," he said. "Many are taking advantage of a system with loopholes in it."

He added that if the migrants truly did fear returning to their home countries, they would have sought asylum in Mexico, which some did.

Not all of the caravan members had intended to seek US asylum, and many broke away from the group before reaching the final destination. Mexico offered many of the migrants temporary travel visas that allowed them time to apply for legal immigration status in Mexico, rather than journeying on to the US.

But the 18-year-old man who was allowed to enter on Monday told BuzzFeed earlier that day he had already been preparing for what he was going to tell the US immigration officers.

"Attorneys have told me I have a strong case and that there is a good chance I'll win," he said. "I'm determined to fight my case."