As President Biden grapples with an unwieldy crisis at the southern border, his likely 2024 rival has leveled many criticisms — including some baseless and misleading claims.Former President Donald J. Trump has drawn widespread censure after reprising a line that casts undocumented immigrants as “poisoning the blood of our country.”The remark underscored Mr. Trump’s hard-line approach to immigration, which has been central to his platform since he made his first bid for president in 2015. If elected again, he has vowed to carry out mass deportations and enact other strict policies.He and his Republican rivals have pointed to the surge of migrants at the southern border to make their political case. Some Democrats, too, have been critical of the Biden administration’s approach toward immigration.But even with legitimate lines of attack, Mr. Trump has at times turned to baseless and misleading claims during rallies in recent months.Here’s a fact check.WHAT WAS SAID“I read an article recently in a paper … about a man who runs a mental institution in South America, and by the way they’re coming from all over the world. They’re coming from Africa, from Asia, all over, but this happened to be in South America. And he was sitting, the picture was — sitting, reading a newspaper, sort of leisurely, and they were asking him, what are you doing? He goes, I was very busy all my life. I was very proud. I worked 24 hours a day. I was so busy all the time. But now I’m in this mental institution — where he’s been for years — and I’m in the mental institution and I worked very hard on my patients but now we don’t have any patients. They’ve all been brought to the United States.”— during a rally in Nevada this monthThis lacks evidence. Mr. Trump has repeatedly claimed that immigrants crossing the border are coming from “mental institutions” and jails. This particular story would seem to offer specific facts behind that assertion, but there is no evidence that such a report exists.The New York Times could not find any such news account from the start of Mr. Biden’s tenure in January 2021 to March, when Mr. Trump told the same story at a Texas rally.The Trump campaign did not respond when repeatedly asked about the source of this claim. But pressed this year by CNN for factual support for the tale, the campaign provided links that did not corroborate it.Likewise, there is no support for Mr. Trump’s broader claim that countries are “dumping” their prisoners and psychiatric patients in the United States.“We are unaware of any effort by any country or other jurisdiction to empty its mental-health institutions or its jails and prisons to send people with mental-health issues or criminals to the U.S.,” Michelle Mittelstadt, a spokeswoman for the nonpartisan research organization Migration Policy Institute, said in an email.The claim evokes elements of a mass exodus that occurred more than 40 years ago in Cuba, Ms. Mittelstadt noted: the Mariel boatlift of 1980. Some 125,000 people fled to the United States, including inmates from jails and patients from mental health institutions freed by the Cuban leader Fidel Castro.“But there has been no present-day effort by any country, to our knowledge, or any credible reporting by media or others that anything of the like is taking place,” Ms. Mittelstadt said.WHAT WAS SAID“They’ve allowed, I believe, 15 million people into the country from all of these different places like jails, mental institutions, and wait till you see what’s going to happen with all those people.”— during a rally in October in New HampshireThis lacks evidence. Setting aside the baseless suggestion that all undocumented immigrants entering the country are coming from jails and mental institutions, Mr. Trump’s estimate of 15 million is not supported by the data.Customs and Border Protection data shows that U.S. officials recorded nearly eight million encounters at its borders from February 2021, the first full month of Mr. Biden’s presidency, to October 2023.But even then, “encounter does not mean admittance,” Tom Wong, an associate professor of political science and director of the U.S. Immigration Policy Center at the University of California, San Diego, said in an email. “In fact, most encounters lead to expulsions.”For example, C.B.P. data shows that about 2.5 million expulsions occurred under Title 42, a health rule that used the coronavirus as grounds for turning back immigrants illegally crossing the border, from February 2021 until the policy ended in May.Former President Donald J. Trump has at times turned to baseless and misleading claims about immigration in recent months.Rachel Mummey for The New York TimesThe number of encounters also are based on events, not people, and therefore could include the same person more than once.The exact number of people who have entered the country without authorization is hard to pin down because there are also “gotaways” — people who crossed into the country illegally and evaded authorities.But the federal, observational estimates of such people also would not support Mr. Trump’s claim. The secretary of homeland security, Alejandro N. Mayorkas, estimated at a recent hearing that there had been more than 600,000 gotaways in fiscal year 2023, which ended in September. That is also the estimate for fiscal year 2022, according to an inspector general report. And there were more than 391,300 in fiscal year 2021, which began in October 2020 under Mr. Trump and ended in September 2021 under Mr. Biden.In terms of migrants with criminal records, officials encountered nearly 45,000 at ports of entry since the start of fiscal year 2021. Between ports of entry in that period, officials encountered another 40,000 noncitizens with criminal records.While Mr. Trump in this instance claimed the country had allowed 15 million migrants to enter, he has at other times predicted that would be the total figure by the end of Mr. Biden’s term. That would be larger than the estimated total population of unauthorized immigrants living in the United States — about 10.5 million in 2021, according to the Pew Research Center.WHAT WAS SAID“In the past three years, Biden has spent over $1 billion to put up illegal aliens in hotels, some of the most luxurious hotels in the country. Meanwhile, we have 33,000 homeless American veterans. Can you believe it?”— during a rally in November in New HampshireThis needs context. Mr. Trump’s figure of homeless veterans refers to a 2022 estimate by the Department of Housing and Urban Development. That number includes about 19,500 veterans who were in shelters when the count was conducted. And both the 2022 estimate and a new tally for 2023 — which reported nearly 35,600 homeless veterans — are actually down slightly from when Mr. Trump was in office, continuing an overall downward trend since 2009.As for migrant housing, Immigration and Customs Enforcement contracted in 2021 with a nonprofit group to house border arrivals at a handful of hotels in Texas and Arizona, as a 2022 homeland security inspector general report details. The contract totaled more than $130 million and ended in 2022. The Trump administration also turned to hotels in 2020 to hold migrant children and families before expelling them.The Biden administration has not directly spent $1 billion to place immigrants in hotels. But cities are indeed facing steep costs for sheltering and caring for border arrivals — including through hotels. The Trump campaign did not indicate where Mr. Trump had obtained the $1 billion figure, but it is possible he was referring to a federal initiative that provides funding to local governments and nongovernment groups to help offset those costs.The program was in fact first authorized through 2019 legislation signed by Mr. Trump. While it allows nonfederal entities to seek grants for housing migrants in hotels and motels, it is not exclusive to that. Congress provided the program $110 million in fiscal year 2021 and $150 million in fiscal year 2022.Lawmakers recently replaced the initiative with a new shelter and services program. For fiscal year 2023, officials earmarked $425 million for the old program and $363.8 million for the new one.All told, the federal government has allocated about $1 billion since fiscal year 2021, which includes the last few months under the Trump administration, toward local efforts to feed and shelter migrants around the country — not only hotel expenses.While FEMA discloses recipients of the funding, it does not say how much each grant is used specifically on hotel costs.WHAT WAS SAID“We cannot forget that the same people that attacked Israel are right now pouring in at levels that nobody can believe into our beautiful U.S.A. through our totally open border.”— during a rally in Iowa in OctoberThis lacks evidence. Mr. Trump offered no evidence that people affiliated with Hamas, the militant group that staged a brutal assault on Israel in early October, are “pouring” into the country at record levels. And experts say they are unaware of data that would support that contention.If the former president’s statement was meant to convey that terrorists more generally are “pouring in” at the border, he could be referring to the rising number of encounters at the southern border with people on a terrorism watch list. The list includes known and suspected terrorists as well as people affiliated with them.A total of 169 noncitizens on that list tried to illegally enter the United States at the southern border in fiscal year 2023, which ended in September, up from three in fiscal year 2020, according to C.B.P. statistics.Still, it is unclear what that says about the terrorism threat, said Alex Nowrasteh, vice president for economic and social policy studies at the libertarian Cato Institute. There is no record of a terrorist attack being committed on American soil by an immigrant who crossed the southern border illegally. (In 2008, three brothers who had come to the United States illegally years earlier as children, from Yugoslavia, were convicted of conspiring to kill American soldiers at Fort Dix in New Jersey.)Apprehended individuals on the list are supposed to remain in government custody as they await removal proceedings, Mr. Nowrasteh said.Curious about the accuracy of a claim? Email factcheck@nytimes.com. More