President Donald Trump says the US "will now begin cutting off or substantially reducing" the amount of foreign aid given to Guatemala, Honduras, and El Salvador, saying they were "not able to do the job" of stopping migrants from leaving their countries and "coming illegally" to the US.

His Twitter comments Monday came as a group of several thousand migrants, mostly from Honduras, spent Sunday night in the southern Mexican city of Tapachula as they continued their trek toward the United States and away from what they say is untenable violence and poverty at home.

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Trump said, "Sadly, it looks like Mexico's Police and Military are unable to stop the Caravan heading to the Southern Border of the United States. Criminals and unknown Middle Easterners are mixed in. I have alerted Border Patrol and Military that this is a National Emergy (sic). Must change laws!"

Two weeks ahead of nationwide congressional elections in the US, the US leader, a Republican, laid the blame for the latest mass migration toward the southern US border on opposition Democrats.

"Every time you see a Caravan, or people illegally coming, or attempting to come, into our Country illegally," Trump said, "think of and blame the Democrats for not giving us the votes to change our pathetic Immigration Laws! Remember the Midterms! So unfair to those who come in legally."

The United Nations refugee agency has 32 workers in Mexico to provide humanitarian assistance and legal advice to the migrants, with its local partners offering asylum information to those who want to stay. The International Organization for Migration said large numbers of migrants arrived in Mexico on Monday, with many likely to remain for an extended period. IOM estimates that more than 7,200 people are in the caravan, with many of them planning to continue their march northward.

Secretary of State Mike Pompeo said the Trump administration is closely following the progress of the caravan and accused the migrants of violating the sovereignty, laws and procedures of the countries they are traveling through.

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Pompeo said the United States is concerned about violence provoked by some members of the caravan and "the apparent political motivation of some organizers" without giving specifics about either.

He also pledged US support for Mexican government efforts to address refugee and migration issues.

Authorities in southern Mexico largely left the migrants alone Sunday as they walked toward the day's destination in Chiapas state.

The Mexican government has pledged to process asylum requests for migrants who apply. The country's interior ministry reported that on Friday, Saturday and Sunday a total 1,028 people had requested refugee status.

Mexico's National Migration Institute said it reiterates its duty to safeguard the human rights of migrants who enter its territory.

Trump has threatened to deploy the US military to stop what he calls an "assault on our country at the Southern Border," and portrayed the migrant group as containing "many criminals."

"Full efforts are being made to stop the onslaught of illegal aliens from crossing our Southern Border," he tweeted Sunday. "People have to apply for asylum in Mexico first, and if they fail to do that, the US will turn them away. The courts are asking the US to do things that are not doable!"

Trump blames the opposition Democrats for what he says are immigration laws and border policies that are not strong enough to protect the United States.

"The Caravans are a disgrace to the Democrat Party. Change the immigration laws NOW!" he tweeted.

Pueblo Sin Fronteras, an organization that helps the migrant caravans in Central America, says governments in the region have adopted "a policy of fear and racism imposed by the United States" and are not considering the reasons why people are seeking somewhere new to go.

"They are walking in mass exodus because they cannot live in their country anymore due to extreme violence, lack of opportunity, and the corruption and impunity that has expelled them from their homes," the group said in a statement Sunday.

Mexico's incoming president Andres Manuel Lopez Obrador told supporters at a rally Sunday in Chiapas that he would be sending a letter to Trump proposing Mexico, the United States and Canada work together to invest in development in Central America to address poverty.

Lopez Obrador, who takes office Dec. 1, said people who leave their home do so not because they want to, but out of necessity.

He has pledged to offer migrants work visas, and said Sunday that Mexico has to guarantee human rights and that above all, the migrant families, women and children will have protection.

Read More: 8 "Caravan" Migrants Admitted to US to Make Cases for Asylum

"Nothing bad will happen to the Central American migrants," Lopez Obrador said.

Aid group Save the Children expressed concern Sunday about children who were sleeping outside in Tapachula and Suchiate either because places were full or the children feared they would be detained once inside.

The group estimates 1 in 4 members of the caravan are children.

The Red Cross said Saturday that many of the people it is helping along the caravan route, a majority of them women and children, "are suffering from dehydration, stomach infections, and foot injuries as they walk the long journey."

Walter Cotte, International Federation of Red Cross and Red Crescent Societies regional director for the Americas, said in a statement: "It is imperative that the dignity and security of families are safeguarded and they are kept together."

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Trump Admin Might Cut Aid to Guatemala, Honduras, El Salvador Over Migrant Caravan