SAN DIEGO (AP) — A group of Central Americans who journeyed in a caravan to the U.S. border resolved to turn themselves in and ask for asylum Sunday in a direct challenge to the Trump administration — only to have U.S. immigration officials announce that the San Diego crossing was already at capacity.

Nearly 200 migrants, many traveling with children, had decided to apply for protection at the nation's busiest border crossing after many fled violence in their home countries, organizers said. The caravan got attention after President Donald Trump and members of his Cabinet called it a threat to the United States.

Shortly before the migrants were expected to arrive, U.S. Customs and Border Protection said San Diego's San Ysidro crossing would not immediately be able to handle more asylum seekers. It can hold about 300 people at a time, and officials had been warning that it might fill up.

"At this time, we have reached capacity at the San Ysidro port of entry for CBP officers to be able to bring additional persons traveling without appropriate entry documentation into the port of entry for processing," Commissioner Kevin McAleenan said in a statement. "Those individuals may need to wait in Mexico as CBP officers work to process those already within our facilities."

He said the crossing could take in additional people as space and resources become available. Despite the news, about 200 migrants still started walking toward the port.

Rodulfo Figueroa, the top Mexican immigration official in Baja California state, told caravan organizers to send in an initial group of 20 migrants to see if U.S. border inspectors would entertain their request for asylum.

Figueroa said he doesn't know if they would be allowed in and had not received word from U.S. immigration officials.

Nicole Ramos, an attorney working on behalf of caravan members, expressed disbelief that U.S. authorities cannot process more asylum seekers until its backlog eases.

"They have been well aware that a caravan is going to arrive at the border," she said at a news conference. "The failure to prepare and failure to get sufficient agents and resources is not the fault of the most vulnerable among us. We can build a base in Iraq in under a week. We can't process 200 refugees. I don't believe it."

The migrants had made their way north by foot, freight train and bus over the past month, many of them saying they feared for their lives in their home countries.

The Trump administration has been tracking the caravan since it started in Mexico on March 25 near the Guatemala border. Attorney General Jeff Sessions has called the caravan "a deliberate attempt to undermine our laws and overwhelm our system."

Administration officials have railed against what they call America's "catch and release" policies that allow people requesting asylum to be released from custody into the U.S. while their claims make their way through the courts, a process that can last a year.

Wendi Yaneri Garcia said she is confident she will be released while her asylum case is pending because she is traveling alone with her 2-year-old son, who has been sick.

"All I want is a place where I can work and raise my son," the 36-year-old said.

She said that police in her hometown of Atlantida, Honduras, jailed her for protesting construction of a hydroelectric plant and that she received death threats after being released.

Nefi Hernandez, 24, said a gang in his hometown of San Pedro Sula, Honduras, threatened to kill him and his family if he did not sell drugs. He intended to seek asylum with his wife and baby daughter, who was born on the journey through Mexico.

Jose Cazares, 31, said he faced death threats in the Honduran city of Yoro because a gang member suspected of killing the mother of his children learned one of his sons reported the crime to police.

Earlier Sunday, the migrants boarded five old school buses to attend a rally at a Pacific Ocean beach, with supporters gathering on both sides of the border fence and some climbing the barrier to sit or to wave signs.

The travelers face an uncertain future if they ask for asylum. U.S. immigration lawyers warned them that they face possible separation from their children and detention for many months.