Meanwhile, several migrants swam around or climbed over the border barrier overnight and were detained by USA officials.
Authorities in the Mexican city of Tijuana said Friday they have begun moving Central American migrants from an overcrowded shelter on the border to an events hall further away.
The department announced Friday that the man, who was identified as 30-year-old Wuilson Lazo-Ramos, was arrested near Ajo, Arizona.
Migrants would be allowed to stay - many who hope to cross don't want to move far away from the border - but they will have to find their own food, Magallanes said.
The Border Patrol office in San Diego said via Twitter that pedestrian crossings have been suspended at the San Ysidro port of entry at both the East and West facilities.
"The truth is that we were told that if we didn't go to the caravan, then we were not with them, and you know, since we departed from Honduras, we came in a caravan", she told the Epoch Times.
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The migrants were part of a caravan of about 7,000 people that arrived in Tijuana two weeks ago. That prompted Border Patrol agents to launch tear gas and pepper spray balls to quell the unrest.
One Honduran man tried to swim to the USA side but quickly got in trouble in the rough waters of the Pacific.
Thousands of caravan members have been arriving in recent week, almost 6,000 people have been staying at a sports complex in Tijuana packed into a space adequate for half that many people.
One of those men had been previously deported.
The other adults were not prosecuted because Customs and Border Protection didn't have enough information to pursue charges, including the name of the arresting officers, according to the official, who said it was a chaotic scene.
The spokesman said there are three confirmed cases of tuberculosis, four cases of HIV/AIDS, and four cases of chickenpox, in addition to at least 101 cases of lice and multiple skin infections. About 1,000 tried to rush the border and illegally cross into the U.S., Border Patrol chief Carla Provost said. Judge Jon Tigar rejected the Justice Department's request to suspend his earlier order temporarily blocking the ban.