An official from Mexico's National Institute of Migration said Mexico plans to disperse the caravan by Wednesday, but some people such as pregnant women or those with disabilities would get humanitarian visas.
"It's not because of Donald Trump", Irineo Mujica, director of the advocacy group, told Reuters on Wednesday.
Mr. Trump has been fixated on the caravan this week, making it a recurring motif in his tweets, warning that it represented a grave threat to the sovereignty of the United States, and using it as a justification for sending the military to reinforce the southwest border. Trump reacted by threatening to end the North American Free Trade Agreement (NAFTA) and also end foreign aid to Honduras, where numerous migrants originated, if the caravan was not stopped.
"All they want is a place to live in peace, where they can work without having guns pointed at them, without being forced to join a gang", said Mujica.
The video was Peña-Nieto's first public response to Trump since the USA president kicked off a week-long barrage of tweets and statements blasting everything from US immigration laws, the Mexican government, Democrats, NAFTA and Honduran asylum-seekers.
'We do not accept the construction of a wall on our border.
An organizer of the volunteer group that organized the caravan, said that migrants would keep moving, but in smaller groups according to a report Tuesday.
For its part, Mexico was also caught off balance by the furor over the caravan.
"We've recently seen the numbers of illegal border crossings rise from 40-year lows last April to back to previous levels", Department of Homeland Security Secretary Kirstjen Nielsen said.
Monday evening, the Mexican government started to break up a migrant caravan that was moving through southern Mexico.
Mexican officials privately say they hope the offer will entice enough of the migrants to apply for asylum here to make the crisis go away.