By Jonathan Allen and Mica Rosenberg

(Reuters) - New York Mayor Eric Adams this weekend visits the Darien Gap, a treacherous rainforest traversed by thousands of mostly South American migrants a month, to try to dissuade them from coming as U.S. cities like his own struggle to accommodate them.

His unusual trip to Mexico, Ecuador and Colombia underscores the criticism that he and other Democratic city and state leaders have voiced for the immigration policies of their fellow Democrat, President Joe Biden.

"Our hearts are endless, but our resources are not," Adams said at a press conference during a stop in Puebla in central Mexico. "They deserve a more dignified environment than we are able to give."

That New York City's mayor has traveled thousands of miles to make his case highlights how the latest wave of migrants is reshaping the immigration debate among some Democratic leaders.

Republican governors near the border have been fiercely critical of Biden's border policies. Texas Governor Greg Abbott has been busing thousands of migrants north to New York, Chicago and other Democratic-controlled cities.

But in recent months, Adams and some other Democrats have said their cities far from the southern border are now also being strained by a surge of new arrivals, many of whom are fleeing violence and economic distress in Venezuela.

Illinois Governor J.B. Pritzker, a Democrat, called the influx "untenable" in a letter to Biden on Monday, while the mayor of Chicago, Brandon Johnson, also a Democrat, said this week he planned to visit the U.S.-Mexico border after more than 17,000 migrants had arrived in the city since last year.

Immigration will likely be a campaign theme in the U.S. 2024 presidential race with a 54% majority of Americans agreeing with the statement that "immigration is making life harder for native-born Americans," a September Reuters/Ipsos poll found.

Some 73% of Republicans and 37% of Democrats surveyed agreed with that statement.

Biden's administration said on Thursday it will add sections to a wall to reduce the record number of migrant crossings at the U.S.-Mexico border, carrying forward a signature policy of former President Donald Trump, Biden's predecessor and the frontrunner in the Republican race to challenge him in 2024.

ADAMS SAYS ASYLUM SEEKERS ARE MISLED

Adams has said that many asylum seekers set out for New York in the belief they would receive jobs and rooms in "five-star hotels" upon arrival, led astray by misinformation spread by smugglers and on social media. "We're saying that is not true."

The New York Immigration Coalition and other advocacy groups have criticized Adams' trip as an ineffectual use of time and money when Adams is slashing city budgets, saying New York's mayor has no influence on international policy and that the city has mismanaged the existing shelter system. They have said his remarks last month that the migrant crisis will "destroy New York City" are inflammatory.

This week, Adams' office asked a judge to end a decades-old right-to-shelter mandate, which requires New York to provide a place to sleep to anyone who needs one. The city says hundreds of migrants are arriving each day with nowhere to stay.

The effort is opposed by the Legal Aid Society, which advocates for newly arrived migrants seeking shelter.

Joshua Goldfein, a Legal Aid attorney, said it was unclear whether physically traveling to the Darien Gap might help with the crisis in New York City, and there was more the mayor could be doing without leaving his hometown.

Much of the misinformation the mayor has described was being seen by asylum seekers on cellphones, in TikTok videos being shared on WhatsApp. To counter this, the city should be making its own videos in the languages of the migrants traveling north, Goldfein said.

"All three levels of government need to step up here to solve this problem," Goldfein said: the federal government should be giving people work authorization, and assisting local officials in states and cities to meet new arrivals and help them resettle in less expensive and strained cities. "But they're all sort of pointing fingers at each other."

Adams will tour migrant shelters in Ecuador on Friday, and on Saturday he will visit the Darien Gap, a roadless jungle region straddling the Colombia-Panama border that requires several days of dangerous hiking to cross, although it is unclear how far in the mayor will go.