The caravan resumed its journey Monday in the state of Chiapas in southern Mexico, setting out from Tapachula, near the border, for the town of Huixtla, around 40 kilometers (25 miles) away.
Under a proposed bilateral agreement that the Trump administration has discussed with Mexican officials, United States border officials would be able to legally turn back asylum-seekers who first pass through Mexico, forcing them to seek protection south of the border.
In April, Mexican immigration officials had some success in dispersing the smaller caravan by processing many who chose to seek refugee status in Mexico, but some did continue on to the USA border where they were processed over several days.
President Trump last week warned Mexico that "we're calling up the military - not the Guard" if the government can not stop the caravan from reaching the border.
Speaking on the Televisa network, Videgaray did not seem concerned about Trump's threat to close the U.S. -Mexico border, saying it had to be viewed in light of the hotly contested U.S. midterm elections, in which Trump has made border security a major campaign issue.
He predicted that, like the caravan in April, Mexico will respond with measures like granting asylum to some migrants who qualify while deporting others who don't, perhaps not eliminating the caravan entirely but significantly reducing its size before it reaches the USA border.
Local news reports said once the migrants broke into Mexico, authorities there did not try to stop them.
"I am asking President Trump to help us.people like me.who want to survive".
Here, some of them are pushed by others after breaking a fence on the Guatemalan side of the border bridge.
"We'll continue this operation for as long as is necessary", Hernandez said in a post on Twitter.
Cristian, a 34-year-old cellphone repairman from San Pedro Sula, said he left Honduras because gang members had demanded protection payments of $83 a month, a fifth of his income.
Violent clashes erupted on Friday when thousands rushed across the bridge, and were met by Mexican police officers armed with riot gear and pepper spray.
"We chose to join those who were going", said Edgar Aguilar.
But American President Donald Trump has made clear that they are not welcome and that he may call in soldiers to stop them marching into the States. Later, he retweeted a video of Mexican federal police arriving at the Guatemalan border and wrote: "Thank you Mexico, we look forward to working with you!"
"We shall work with our partners in the region to investigate and prosecute to the fullest extent of the law all who seek to encourage and profit from irregular migration", she said.
Lopez Obrador, who takes office December 1, ran on a pro-immigration platform, promising jobs and work visas in Mexico to Central American migrants. Mexican Federal Police, however, did not stop those migrants who crossed the river into Mexico by swimming or by using a raft.
The Mexican Interior Ministry said on Saturday that 640 Honduran migrants have requested refuge in Mexico.
Just over two weeks before the US midterm elections, in which Trump has made illegal immigration a rallying call for his Republican base, Trump again blasted Democrats for what he called "pathetic" immigration laws.
Even though the goal is to reach the USA border, she said: "We only want to work and if a job turns up in Mexico, I would do it".
And according to worldwide law, the USA can not deport asylum seekers without first determining the validity of their claim.
"So as of this moment, I thank Mexico", Trump said Friday at an event in Scottsdale, Arizona.
Ulises Garcia, a Red Cross official, said some migrants with injuries from their hard trek refused to be taken to clinics or hospitals, because they didn't want to leave the caravan.
Mr. Trump, who has for days now blasted Democrats and the United States' immigration laws over the caravan, also warned, without citing evidence, that "unknown Middle Easterners" and "criminals" are mixed into the caravan of people heading towards the United States' southern border.
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