Their journey of almost 4500km - mostly on foot - began in San Pedro Sula, Honduras, last Friday. That would be about one fifth of his income.
Police and immigration agents began letting small groups of 10, 20 or 30 people through the gates if they wanted to apply for refugee status.
The president appears to agree, promising to deploy the military if the caravan makes it to the US border.
"Honestly, I want to get to the states to contribute to that country, to do any kind of work, picking up garbage", he said.
But many became impatient and circumvented the border gate, crossing the river on rafts, by swimming or by wading in full view of the hundreds of Mexican police manning the blockade on the bridge.
Addressing the caravan situation, Pompeo said, "The Mexican government is making all the decisions on how to address this".
Some migrants collapsed, coughing or weeping, according video from the scene.
"I am watching the Democrat Party led (because they want Open Borders and existing weak laws) assault on our country by Guatemala, Honduras and El Salvador, whose leaders are doing little to stop this large flow of people, INCLUDING MANY CRIMINALS, from entering Mexico to U.S.", Trump tweeted Thursday morning.
But even as the US -bound caravan moved into Mexico, Mexican police in riot gear were deployed at highway junctions, as a military helicopter circled overhead, prompting many migrants to wonder if police will make mass arrests and seek to turn the caravan back. About 50 managed to push their way through before officers unleashed pepper spray and the rest retreated, joining the sea of humanity on the bridge. "No one will stop us, only God", he said. Men and women, some with young children and babies drenched in sweat, began storming and climbing the barrier - tearing it down.
Edwin Santos of San Pedro Sula was one of the first to race past helpless Guatemalan police, clutching the hands of his father and wife. He should be happy because we're going to work.
"I don't know what happened, I thought we were going to cross peacefully and then suddenly there were rocks flying and tear gas", she told AFP.
Mr Hernandez said another 486 migrants were in transit back to Honduras, a roughly 12-hour trip by road.
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Several thousand Honduran migrants seeking to escape violence and poverty moved through Guatemala on the way to Mexico, with some hoping to enter the United States.
"There is nothing there", Orellana said.
After an emergency meeting in Guatemala, the presidents of Honduras and Guatemala said an estimated 5,400 migrants have entered Guatemala since the caravan was announced a week ago and about 2,000 people have returned voluntarily to Honduras. If they are once again told they will need to wait on the Mexican side for days or even weeks while their cases are handled, we've already seen what their answer will be.
Where easily 3,000 people were on the bridge the previous day, the crowd had thinned out considerably by Saturday. "It's the third time that I'm trying to cross", the 22-year-old Salvadoran said.
"They're not coming into this country", Trump said at the rally.
USA officials - and President Trump - warned the marchers when they started the march earlier this month in Honduras.
Earlier this week, President Trump tweeted that he will send USA troops to close the border with Mexico if the group of migrants is not stopped.
On Thursday, the government of Mexico dispatched two Boeing 727s filled with police and military personnel to the country's southern border with Guatemala in preparation of preserving law and order upon the caravan's arrival.
UNHCR spokesman Charlie Yaxley said the agency is reinforcing capacity in southern Mexico to offer counseling, legal assistance and humanitarian aid to asylum-seekers.
Trump has pointed to the caravan as a key issue in the upcoming election, describing the group's approach as an assault on the U.S. border.