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 US southern border via a large migrant caravan - while also describing such caravans as "a disgrace to the Democrat Party".
In Mexico City, US Secretary of State Mike Pompeo said after meeting his Mexican counterpart that migration across the two countries' border is reaching "a moment of crisis" and posing "a challenge for American sovereignty".
Most of the migrants are Hondurans who cite widespread poverty and gang violence as their reasons for wanting to leave their country.
Numerous migrants, the overwhelming majority fleeing violence and poverty in Honduras, slept on the bridge through heavy rain overnight, dozens of them crammed against a metal border gate guarded by Mexican police.
"Every time there's a (migrant) caravan there are police sent to the southern border. but we've never seen anything as dramatic as we're seeing today", said Ms Eunice Rendon, coordinator of migrant advocacy group Agenda Migrante. I hope they continue, but as of this moment I thank Mexico. They were taken safely on a raft to Mexican territory.
An estimated 4,000 people made up the group waiting to cross into Mexico this week - a reflection of what Elizabeth Oglesby, a professor at the University of Arizona's Center for Latin American Studies, described as the growing size of migrant caravans.
President Trump threatened in a series of tweets on Thursday to "call up the US military and close our SOUTHERN BORDER" if Mexico doesn't do anything to stop the flow of migrants moving north.
"A lot of people say 'I wonder who started that caravan?'" he said.
"Let's all walk together!" and "Yes, we can!" they said.
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Mexico's government has said throughout the past week that it would register the migrants and process requests for asylum.
"We respect the freedom of movement of immigrants and above all, based on Mexican immigration law, we are receiving people who are choosing to submit to immigration in order to submit an application for refugee or a humanitarian visa, depending on the case", Mr Moreno said. The situation, however, has changed in recent months as the Trump administration takes a strong hand in pressuring Mexico to crack down on illegal border crossings.
But many became impatient and circumventing the border gate, crossing the river on rafts, by swimming or by wading in full view of the hundreds of Mexican police manning the blockade on the bridge.
The Mexican government also released handout video of migrants on a bus being told by an official that they would be assisted in being processed for any asylum claims they may have.
"This is an organized effort to come through and violate the sovereignty of Mexico", Pompeo said. God is going to soften Trump's hard heart.
Mr Trump has also threatened to cut aid to countries allowing the caravan to pass. Trump said on Twitter. "These aren't babies, these aren't little angels coming into our country", he said Friday at the White House.
"If that doesn't work out, we're calling up the military - not the (National) Guard - we're calling up the military", he told reporters.
The first members of the 3,000-strong caravan began arriving in the Guatemalan border town of Tecun Uman on buses and trucks early Thursday, but the bulk of the group sloshed into town on foot in a downpour late in the afternoon and into the evening.
A family in the migrant caravan waits at a gate separating Guatemala from Mexico.