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The Mexican government said Monday night, if members of the \"caravan\" reached any U.S. ports of entry, it would be up to American authorities to decide whether to let them enter the country.

It was not the job of the Mexican government, it said, \"to exercise immigration decisions of the United States or any other nation, so it will be the competent authorities of the United States that decide, if appropriate, to authorize or not the entry to its territory of the members of the caravan who request it through the authorized entry and exit ports.\"

The march, versions of which have happened on and off since 2010 -- including twice last year -- will conclude at an official port of entry in California, Arizona or New Mexico, depending on train schedules, Mensing told ABC News this weekend.

Also late Monday, the organizers, Pueblo Sin Fronteras, said it continued to stand with \"displaced people\" despite Trump's \"bullying.\"

It said the newfound attention on the march has led \"right-wing white surpremacists\" to threaten and call for violence against the marchers and its supporters, including \"personally with hate-filled phone calls and emails\" and calls to open fire and carry out drone strikes on them.

\"On April 1, the U.S. President opportunistically invoked refugee caravans as a pretext for threatening immigrants already in the country, specifically DACA recipients, with a so-called 'Nuclear Option' to remove their protections from detention and deportation,\" Pueblo Sin Fronteras said in its statement, referring to a legislative move Trump has backed. \"Trump is trying to turn Central American refugees and other immigrant communities against each other, and use them as a bargaining chip with Mexico.\"

\"In the face of this bullying and these threats of mass violence,\" the group continued, \"we continue to stand in solidarity with displaced people of all races, ethnicities, creeds, abilities, and gender and sexual identities. We invite all those who envision a world where human rights are honored to join us.\"

ABC News' Joshua Hoyos contributed to this report.

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