The ease of entering is a selling point for the cartels, who control the drug and human smuggling routes into the US and require payment of a "mafia fee" just to cross the U.S.
UNHCR spokesman Charlie Yaxley said the agency is reinforcing capacity in southern Mexico to offer counseling, legal assistance and humanitarian aid to asylum-seekers.
Police deployed pepper spray after some migrants tried to push their way through the Mexican side.
They camped out on the bridge last night with thousands of the other migrants.
Mexico's government vowed on Friday to meet the challenge of a caravan of Central American migrants heading north, after U.S. President Donald Trump threatened to shut down the U.S-Mexico border to halt its passage. Authorities began handing out numbers for people to be processed in a strategy seen before at USA border posts when dealing with large numbers of migrants.
"It is a blessing that they have given us food", Martinez said.
Thousands of Central Americans in a USA -bound migrant caravan began pushing their way from Guatemala into Mexico on Friday, pulling down border fences and storming toward the Mexican immigration post.
"Tomorrow we go home", he said.
The migrants could be seen passing babies overhead through the crowd, as women holding crying children by the hand or pressing their infants to their chests streamed past the broken metal barriers and onto the bridge.
"We are going to the United States!" he said. "Nobody is going to stop us!" Others have been allowed through in groups by the Mexican border officials and are applying for asylum or other forms of legal residence.
Gerardo Hernandez, head of the local Mexican government's emergency services, confirmed to Reuters that more than 5,100 migrants were registered in three shelters in the Mexican border town of Ciudad Hidalgo. "This is an exodus", said Ruben Figuerora of the Mesoamerican Migrant Movement, standing on the muddy bank of the Mexican side of the Suchite river, as a line of young male migrants walked past him after crossing on a raft.
Mexico has always been the major route for Central Americans seeking to make it to the United States.
"There is nothing there", Orellana said.
But "they don't all register here for fear of being detained", admitted Hernandez. Outside Tapachula about 500 federal police briefly gathered along the highway on buses and in patrol units, but officers said their instructions were to maintain traffic on the highway not stop the caravan.
"On Thursdays I paid the 18th Street gang, and on Saturday the MS-13", Aguilar said.
"Mexico does not and will not allow (people) to enter its territory illegally, let alone violently", he said.
Pompeo said he had a "good conversation" with Mexican President Enrique Peña Nieto and that the two countries are jointly focused on the caravan issue, but "the Mexican government is making all the decisions on how to address this".
Trump has made it clear to Mexico that he is monitoring its response.
There are fears among Trump supporters that the U.S. president might allow protection for so-called Dreamers - a group of hundreds of thousands of young immigrants to the United States - in exchange for an estimated $25 billion required to construct a border wall with Mexico, the Hill reported.
They are part of a so-called caravan of folks who hoped to get to the U.S. border and eventually into the U.S. Reporter Emily Green has been traveling with them, and she's now on the bridge that separates Guatemala from Mexico. Her husband, David Martinez, said they were exhausted, but had to push on to reach their goal of making it to the U.S.
MARTIN: But you said that there are still several thousand people there, so that suggests that most of the people are going to try to stay and move on, keep moving with the caravan.