(CNN) - Thousands of Central American migrants fleeing poverty and violence packed a bridge connecting Guatemala and Mexico in sweltering heat Saturday as part of a politically charged, US-bound caravan.
At least one migrant, from Honduras, was killed after falling from a vehicle, Honduran President Juan Orlando Hernandez told reporters after meeting with his Guatemalan counterpart.
"What we have seen in recent days is without precedent," he said.
At one point Saturday, several dozen migrants on the desperate trek north sang the Honduran national anthem on the edge of the muddy Suchiate River between Tecun Uman, Guatemala, and Ciudad Hidalgo, Mexico.
Others, including a middle-aged woman and her three children, lowered themselves from the bridge on a ladder and climbed onto a makeshift raft that delivered them to Mexico as federal police officers watched nearby.
Exhausted migrants waded across the Suchiate, holding their belongings above the chest-high waters. Rafts ferried others to Mexico.
A group of men helping the migrants across the river chanted "Si se puede" in Spanish, or "Yes you can." Occasionally, young men jumped from the bridge into the river as onlookers cheered.
Once in Mexico, the migrants rested at a nearby warehouse that turned into a shelter, where they received food, water and medical care.
Their journey continued one day after defiant caravan members rushed past border gates only to be stopped by rows of Mexican riot police who dispersed tear gas and smoke canisters into the crowds.
The slow procession north has led President Donald Trump to threaten to cut aid to Central American nations and to send troops to the US border if Mexico fails to stop the surge.
After US Secretary of State Mike Pompeo and top officials met in the Mexican capital the night before, State Department spokeswoman Heather Nauert said Saturday that Mexico was "fully engaged in finding a solution that encourages safe, secure and orderly migration."
At a rally in Elko, Nevada, Trump attacked Democrats, suggesting they want more caravans. He called the migrant situation "sad from both sides."
On the border between Mexico and Guatemala, members of the caravan seemed undeterred by rhetoric from Washington.
"Our message is we're not criminals," Honduran migrant Orlean Herrera said. "We're coming over here because we need a better life. That's why we're here."
Another man shouted, "Donald Trump is the anti-Christ."
By Saturday morning, the bridge was teeming with men, women, teenagers and children. Some fainted from the heat.
Mexican authorities starting Friday were allowing a trickle of migrants, starting with women and children, to pass through gates near the bridge to Tapachula, where they boarded buses for refugee camps.
Others, chanting and cheering, pushed through or climbed over a steel gate Friday until riot police repelled them with tear gas and smoke canisters. Video showed coughing and weeping migrants collapsed on the bridge. Several police and an undetermined number of migrants were injured.
It was unclear how many migrants would cross the border and how long that process would take. Mexican authorities have said they'd allow migrants to enter the country if they have valid visas or meet other requirements.
The caravan formed last Saturday in San Pedro Sula, Honduras, and arrived Monday in Guatemala. Migrants told CNN en Español they were bound for the United States, seeking work and fleeing political corruption and violence.
Leaders throughout the region have publicly urged them for days to turn back.
Hernandez, the Honduran President, said more than 550 Honduran nationals were being bused back to Honduras, and would be offered jobs and other aid upon their arrival.
Pompeo: 'This is an organized effort'
Four Mexican police officers were injured in the standoff at the Mexican border, Pompeo said Friday.
Speaking to reporters on a tarmac in Mexico City, where he'd been meeting with leaders about the caravan and other matters, Pompeo accused the caravan of using women and children "as shields as they make their way through."
"This is an organized effort to come through and violate the sovereignty of Mexico," Pompeo said.
Pompeo said he had a "good conversation" with Mexican President Enrique Peña Nieto and that the two countries are jointly focused on the caravan issue, but "the Mexican government is making all the decisions on how to address this."
Peña Nieto called the confrontation a "unique incident," saying the migrants did not follow the protocols that Mexican immigration officials put in place. He added that some migrants "assaulted and even injured" a number of unarmed members of the Mexican police before they could contain the violence.
"Mexico won't allow anyone entering its territory in an erratic way and much less a violent way," he said Friday night in a video statement released on Twitter.
Peña Nieto said Mexican officials would continue welcoming those traveling with the migrant caravan despite Friday's incident.
Trump, who's been criticizing the caravan with fiery tweets all week, reiterated his threat Friday that if Mexico doesn't stop the migrants, he'll take more drastic action.
"If that doesn't work out, we're calling up the military, not the Guard, we're calling up the military, and we're going to have the military stationed," Trump said. "They're not coming into this country. They might as well turn back."
Trump has also threatened to cut foreign aid to Honduras, Guatemala and El Salvador.
Clash on the bridge
Members of the caravan gathered Friday morning in the border city of Tecun Uman. They waited in sweltering heat -- with temperatures in the mid-80s and 70% humidity -- for hours for others to arrive, packing streets for blocks on end.
They streamed through gates on the Guatemalan side of the border around noon Friday, cheering and chanting as they marched. Celebratory air horns blared as they headed toward Mexico's port of entry.
When they reached a padlocked steel fence near the Mexican side of the bridge, the sheer number of migrants allowed them to break through.
Mexican police pushed them back. Hundreds of Mexican Federal Police sealed the border as drones and helicopters hovered above the crowd.
At least a dozen migrants jumped into the river.
Some of the migrants on the bridge said they joined the caravan because they were desperate for work and had no other choice.
Mexican officials have said people seeking asylum will be processed at Mexico's southern border. But it's unclear what will happen next.
Will they reach the US border?
Mexican authorities haven't said whether anyone from the caravan will be permitted to travel to the US border.
They've previously outlined how they planned to respond to the group, stating:
• Anyone with a valid visa will be able to enter and move freely.
• Anyone who wants to be recognized as a refugee or as a beneficiary of "complementary protection measures" must do so individually. Those who do so will be held "at a migratory station" for as many as 45 business days.
• Anyone who enters "in an irregular manner" will be "rescued and subject to an administrative procedure and, where appropriate, will be returned to their country of origin in a safe and orderly manner."
Mexican authorities also have said they're asking for help from the United Nations High Commissioner for Refugees to process migrants seeking refugee status.
This spring, when another caravan of Central American migrants crossed into Mexico, such policies allowed that group to make it to the US border.
The latest caravan comes weeks before high-stakes midterm elections in the United States, in which many Republican candidates have been echoing the President's messaging about boosting border security and cracking down on illegal immigration.
Trump has pointed to the caravan as a key issue in the upcoming elections, describing the group's approach as an assault on the US border. Immigrant rights advocates have accused his administration of trying to create a crisis to motivate his voter base.
Honduran and Guatemalan leaders talk
Hernandez met Saturday in Guatemala with his counterpart, Jimmy Morales, to work on "implementing a strategy and logistics for the return of Honduran migrants," Hernandez said on Twitter.
Hernandez also sought permission from Guatemala to send Civil Protection personnel to assist Honduran migrants inside Guatemala on Friday, according to several posts on his official Twitter account.
In addition, Hernandez requested authorization to "hire ground transportation for those wishing to return, even airlifting special cases of women, children, the elderly and sick," he said, thanking the Guatemalan government for "all the attention given to our Honduran brothers."
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