Late Tuesday, Trump said via Twitter that the U.S. had conveyed the same message to the governments of Honduras, Guatemala and El Salvador, telling them that U.S. aid will stop if they allow migrants to travel from or across their countries with the intent of entering the United States without permission.
"Anybody entering the United States illegally will be arrested and detained, prior to being sent back to their country!" he addee.
Amid the tweeting, the migrants continued their trek. Despite having walked all day Monday with swollen, blistered and aching feet, the group rose shortly after sunrise from sleeping on the ground in their clothes in the town of Esquipulas.
Dozens attended Mass at the basilica in the city just across the border from Honduras and about 90 miles (150 kilometers) east of Guatemala City. The migrants resumed their journey escorted by Guatemalan police and covered some 30 miles to arrive in the town of Chiquimula for the night.
The group's numbers have snowballed since about 160 migrants departed Friday from the Honduran city of San Pedro Sula, with many people joining spontaneously carrying just a few belongings. A Guatemalan priest estimated more than 2,000 were fed at three shelters run by the Roman Catholic Church.
Three weeks before midterm elections in the United States, the caravan elicited a tough response from Trump.
Earlier Tuesday, Trump had aimed his threat at Honduras.
However, the Central American nation's ability to do anything appeared limited as the migrants already crossed into Guatemala on Monday, twice pushing past outnumbered police sent to stop them - first at the border and then at a roadblock outside Esquipulas.
Trump did not follow through on a similar threat to the Central American nation in April over an earlier caravan, which eventually petered out in Mexico.
In a statement, Honduras' Foreign Ministry accused unidentified "political sectors" of organizing the caravan with "false promises" of a transit visa through Mexico and the opportunity to seek asylum in the United States.
It urged the migrants not to let themselves "be used by a movement that is obviously political and seeks to upset governability, stability and peace in Honduras and the United States."
Meanwhile, Mexico's immigration authority sent out a fresh warning late Monday that only those who meet entry requirements would be allowed into the country and each migrant would have to satisfy Mexican migration agents. Hondurans need visas to visit Mexico in most cases.
Still, it remains unclear if Mexico and other governments in the region - many of whose own people are migrants - have the political will to physically halt the determined border-crossers, who are fleeing widespread poverty and violence in one of the world's most murderous countries.
"In Honduras there are no jobs, and the jobs that do exist aren't enough to live on," said Jose Francisco Hernandez, a 32-year-old from Copan state in western Honduras. "We can't go to the city because it is full of gang members, and that is hurting us. We decided to migrate from the country to see if we can find a better life."
Carlos Reyes, 20, said he was attacked a week ago for being gay and dressing in women's clothing.
"Some men were going to kill me. ... They wanted to kill me for who I am," Reyes said.
The migrants hope that traveling en masse affords them protection from robbery, assault and other dangers that plague the journey north.
Many carried only a few belongings in backpacks and bottles of water. Some pushed strollers or carried toddlers on their shoulders. As the day wore on, the crowd splintered into smaller groups as some walked faster and others fell behind.
Nery Jose Maldonado Tejada, a 29-year-old from San Pedro Sula, said he lost both feet in a June 2015 freight train accident in Mexico while trying to make it to the United States.
On Tuesday, he was being pushed in a wheelchair by a friend, his lower legs wrapped in bandages and a green duffel bag on his lap. He was intent on making it to the U.S. this time, he said.
"I know that there they can put a prosthesis on my feet and I will be able to walk," Tejada said. "And to work, because my hands are still good."
On Tuesday, Guatemalan officials detained a former Honduran lawmaker, Bartolo Fuentes, who was traveling with the caravan, along with two other men, according to a migration official. Some Honduran organizations had identified Fuentes as a coordinator or spokesman for the caravan, though the migrants said he was merely accompanying them and helping.
Fuentes was to be taken to a shelter in Guatemala City and then deported, said the official, who agreed to tell about the detention only on condition anonymity because she was not authorized to discuss the case.
Guatemala also closed migration facilities at the Agua Caliente border crossing to prevent the entry of any more Hondurans.
Last week, U.S. Vice President Mike Pence urged leaders in Honduras, El Salvador and Guatemala to persuade their citizens to stay home and avoid the long, risky journey to the United States.
On Tuesday, Pence tweeted that he had spoken with Honduran President Juan Orlando Hernandez.
"Delivered strong message from @POTUS: no more aid if caravan is not stopped. Told him U.S. will not tolerate this blatant disregard for our border & sovereignty."
In the evening, Pence tweeted that that he had spoken with Guatemalan President Jimmy Morales and "made clear our borders & sovereignty must be maintained. We expect our partners to do all they can to assist & appreciate their support. Reiterated @POTUS' message: no more aid if it's not stopped!"
Since 2014, the United States has committed $2.6 billion in aid for Honduras, Guatemala and El Salvador. For 2019, Washington has earmarked $65.7 million in aid to Honduras for security, democracy building, human rights and economic and social development programs.
In Honduras, recriminations flew across the political aisle.
Salvador Nasralla, a former opposition presidential candidate, tweeted that stopping the caravan would only bring "temporary relief" for the United States. "What we need is to fight against the corrupt government," he said.
Fernando Anduray, leader of the governing National Party, claimed opponents of Honduras' president were behind the migrant exodus. "It is a strategy planned and financed by the opposition," he said.
Trump's threat seemed unlikely to dissuade the migrants, some of whom expressed displeasure with Hernandez.
Gabriela Natareno, 27, who was traveling with her 16-year-old cousin, said her president is to blame for continued migration by desperate Hondurans.
"He keeps the country mired in poverty and corruption," she said.
Associated Press writers Freddy Cuevas in Tegucigalpa, Honduras, and Luis Alonso in Washington contributed to this report.
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