Mexico's foreign ministry said the U.S. National Guard troops at the border won't be armed.
Mr. Pena Nieto said Mexico stands ready to engage with the U.S.
Trump announced on Wednesday that he was sending National Guard troops to the southern border.
"Trump's conduct has been permanently and systematically, not only disrespectful, but insulting, based on prejudices and misinformation and making frequent use of threats and blackmail", said Laura Rojas, head of the Mexican Senate's Foreign Relations Committee, Reuters reported.
Homeland Security Secretary Kirstjen Nielsen on Wednesday cited a historic uptick in border crossings in spring months like April and advertising from smuggling groups as reasons for the move, calling it in part "anticipating".
In both operations, about 30,000 members of the National Guard were deployed along the border with similar aims.
In the statement, the Mexican government said that if the deployment results in "militarization of the border, it would seriously damage the bilateral relationship".
It also reiterates the "fundamental importance of respecting the dignity and human rights of the Mexicans in the United States".
Mexican President Enrique Pena Nieto scolded President Trump in a new video Thursday for taking a "threatening" attitude toward his country, and praising the Mexican Senate for condemning Mr. Trump's comments this week about immigration.
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In separate messages, the candidates Ricardo Anaya, Jose Antonio Meade and Andres Manuel Lopez Obrador called for national unity and condemned the deployment of USA troops along the border.
The Senate also insisted on reminding President Trump that NAFTA has been of mutual benefit, underlining "the five million USA jobs derived from the Treaty and the Mexican investments for more than 52,000 million dollars in 6,500 United States companies that have employed more than 120,000 people", the report continues.
"This anti-Mexican policy has worked politically because unfortunately there are conservative sectors in the United States with little information and he knows how to awaken an anti-Mexican sentiment, what's called xenophobia, which can be like racism, hatred of foreigners", Lopez Obrador said.
At its height last week, the caravan consisted of nearly 1,500 people mostly from Central America.
"We will reach Puebla in different buses".
The caravan of migrants originally numbered around 1,500.
Finally, the Mexican government has proceeded to grant "transit visas to the people of the migrant caravan" assuring that it will not continue stopping its passage to the United States "to those who wish to continue on their way to request asylum", informed the spokespersons of Peoples Without Borders to the newspaper La Opinión (in Spanish).
Pena Nieto has twice canceled plans to visit Washington after tense phone calls with Trump in which the Republican president refused to back down from his insistence that Mexico pay for his planned border wall. Zelaya said he, his wife, and their three children are awaiting temporary transit visas that would allow them to continue to the USA border, where they hope to request asylum and join relatives in NY. The government said it deported some 400 people and would offer refugee status to others.
Most of the migrants joined the caravan in hope of reaching the US and escaping from violence plaguing their own countries.