By Delphine Schrank

TAPACHULA, Mexico (Reuters) - A U.S.-bound caravan of thousands of mostly Honduran migrants whom President Donald Trump has declared unwelcome, began pouring into the Mexican border city of Tapachula on Sunday, setting up impromptu camps in public parks under a heavy rain.

Many members of the caravan, exhausted from the hours-long trek on foot from the Guatemalan border, mostly ignored police offers to board buses heading to a migrant shelter because of suspicions they might be deported instead.

The migrants have defied threats by Trump that he will close the U.S.-Mexico border if the caravan advances, as well as warnings from the Mexican government that they risk deportation if they cannot justify seeking asylum in Mexico.

Police in riot gear shadowed the caravan's arrival along a southern highway, but did not impede their journey.

Among the throngs hiking into the center of the city was Roger Pineda, a 16-year-old Honduran.

"I just want to find some food and a place to sleep," he said, explaining he joined the caravan last week with five family members and a group of friends from the violent city of San Pedro Sula.

"I hope Trump allows us to make it to the other side," he said.

In Guatemala, local media reported that around 1,000 migrants were traveling north en route to the Mexican border.

Earlier on Sunday, a column of people marched under a burning sun as a military helicopter circled low overhead. Many migrants said they were fleeing a toxic mix of violence, poverty and corruption in Central America.

Most said they felt safer advancing in a large group.

"We're going to make it, we're going to keep moving so long as they don't stop us," said Honduran Jaffe Borjas, 17, marching alongside a childhood friend at the head of the column that stretched far down the highway to the horizon.

Along the route north to Tapachula, about 25 miles (40 km) northwest of the border, some broke into song.

"If you send us back, we will return!" a large crowd shouted in unison.

"We are not criminals, we are workers!"

Mexican President-elect Andres Manuel Lopez Obrador adopted a sympathetic tone toward the caravan on Sunday, promising to provide people with work permits in a speech to supporters in Tuxtla-Gutierrez, about 180 miles (290 km) north of Tapachula.

"I want to tell them they can count on us," said Lopez Obrador to a smattering of applause, reiterating that he is seeking Trump's support to help fund a development plan that could alleviate poverty in Central America and southern Mexico.

Since the convoy formed last weekend, Trump has threatened to halt aid to Honduras and Guatemala, and potentially close the U.S. border with Mexico with the help of the military if the migrants' march is not stopped.

Mexico's government has said throughout the past week that it would register the migrants and process requests for asylum. Those attempting to skip the process would face deportation, but the size of the caravan will test Mexico, which has sought help from the United Nations to manage the issue.

The Honduran government has blamed the political opposition back home for encouraging the exodus of migrants.

On Saturday, Mexican immigration authorities said they had only allowed some 640 migrants through the official border crossing on a bridge spanning the Suchiate River.

The slow pace prompted thousands to cross the river illegally by raft or swimming, according to local officials and migrant organizers.

(Reporting by Delphine Schrank; Editing by Diane Craft and Grant McCool)