Officials said Friday that one caravan member, a 46-year-old Honduran man convicted of murder, was arrested trying to cross the border illegally.
"This caravan has proven they have a propensity for violence and the American people agree: violence against our frontline law enforcement will not be tolerated", Waldman said.
The Health Department said 6,000 are in the city, while U.S. Department of Homeland Secretary Kirstjen Nielsen said this week that there are 8,500 spread across Tijuana and nearby Mexicali.
"They're not the women and children that you see them putting in the front of the caravan so that is what is photographed all the time", she tells OneNewsNow.
Trump has threatened to "permanently" close the U.S.
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Some migrants - unwilling to wait two months to start an asylum application or unable to pay a $3000-$10,000 smuggler's fee price, have simply been trying to cross into the USA on their own.
Children had been separated from their parents this summer during a period when a zero-tolerance policy was used at the border, but the separations were stopped by Trump following worldwide outrage.
Meanwhile, the collective momentum which brought the migrants safely through Mexico has dissipated, as caravan members face the dawning realisation that they may be stuck here for months - and that they will make their next steps alone.
Eight thousand refugees and migrants from Honduras, El Salvador and Gautamala have poured into the city along the US-Mexico border in less than two weeks. They briefly escaped and were captured.
"I didn't really have much of a plan when I left Honduras: everyone said that we would be able to cross the border - I thought it would be easier", said 29-year-old Yocelyn Alvarado as she picked up her 3-year-old son Fernando to stop him wondering into vast puddle of stinking floodwater and sodden trash. He also ran and was later captured. The suspect then tried to grab the handcuffs from the agent, sparking a tug-of-war that sent both men tumbling down a hill.
Numerous migrants who made the trek to the East Tijuana property, some 7 miles (11 km) from the border, appeared thankful to be out of the muck even if most will sleep on thin mattresses on a cold, hard floor. Mud, lice infestations and respiratory infections are rampant. Majority remain determined to enter the USA, although some have accepted temporary work and asylum from Mexico.