President Trump tweeted on Sunday that border security officials continued their "full efforts ... to stop the onslaught of illegal aliens" attempting to travel to the U.S. southern border via a large migrant caravan — while also describing such caravans as "a disgrace to the Democrat Party."

"People have to apply for asylum in Mexico first, and if they fail to do that, the U.S. will turn them away," Trump tweeted. "The courts are asking the U.S. to do things that are not doable!"

The caravan of Central American migrants has anywhere from 5,000 to 10,000 people — mainly from Honduras — according to estimates.

In a follow-up tweet, Trump stated: "The Caravans are a disgrace to the Democrat Party. Change the immigration laws NOW!"

At his campaign rally in Montana Thursday night, Trump told the crowd the upcoming midterm elections will be "an election of Kavanaugh, the caravan, law and order and common sense. ... Remember, it's gonna be an election of the caravan."

The president, without elaborating further, blamed the Democrats for the caravan on Twitter last week, blaming them for "weak laws."

U.S. and Mexican officials agreed upon a plan to handle the approaching caravan, a senior administration official told Fox News on Thursday. The plan includes Mexico requesting that the United Nations High Commissioner for Refugees (UNHCR) establish shelters along its southern border with Central America, the official said.

UNHCR spokesman Chris McGrath confirmed Saturday the agency has “deployed teams to the Mexican border with Guatemala where they are working closely with Mexican officials . . . so all individuals can be processed and provided with essential services, including access to health services, food and shelter.”

MIGRANT CARAVAN GRABS FOR AMERICAN DREAM, AS IMMIGRANTS FORGE UNIQUE CHALLENGE FOR US, MEXICO

Mexico is currently processing the caravan in small groups, and said some 640 asylum claims have been processed since Friday. The country deployed some 700 federal police to the border and has said it will police the border against rogue caravan members who try to cross illegally.

The asylum process can take from 45 to 90 days. Those who are approved will be eligible for relocation in a third country, potentially Mexico or the United States, but there are no guarantees.

During the lengthy asylum process, applicants are supposed to remain near an immigration station and sign in each week. Once granted, they can move freely in Mexico. Last year, roughly 9,000 Central Americans received asylum in Mexico but the system is severely backlogged.

If their asylum claim is denied, Mexico says they will be deported.

Fox News' William La Jeunesse, Elizabeth Zwirz and John Roberts and The Associated Press contributed to this report.