AMNA NAWAZ: Well, as you just heard, Donald Trump doubled down on unsupported claims that Haitian immigrants are abducting and eating people's pets in the small city of Springfield, Ohio.

William Brangham recently visited Springfield, where he spoke to city officials and members of the Haitian community there.

He joins us now.

William, good to see you.

WILLIAM BRANGHAM: Hi, Amna.

AMNA NAWAZ: So we heard Donald Trump there repeat those assertions to tens of millions of people who watched the debate last night.

As you have been reporting, there is no factual evidence for those claims.

So what is actually happening in Springfield?

WILLIAM BRANGHAM: Three weeks ago, our wonderful producer Mary Fecteau and I were in Springfield doing a story about what is going on there.

And Springfield is like a lot of Midwestern Rust Belt cities.

It had a declining population for a long time, but in recent years, they have had an influx of Haitians moving to that community.

This is part of the wave of people that are fleeing the violence in Haiti.

Tens of thousands of people have left that country as it has fallen into chaos.

And about 12,000 to 15,000 of them have ended up in Springfield, Ohio.

And they're here legally under the Temporary Protected Status program.

But their arrival in this community has been rocky, to say the least.

There was an accident that killed a young boy last year.

But, in addition, if you just grow a city's population by 20 percent in just a few years, that causes strains.

We talked to the mayor about this, Rob Rue.

Here's what he had to say.

ROB RUE, Mayor of Springfield, Ohio: The infrastructure of the city, our safety forces, our hospitals, our schools.

Springfield is a close community and has a big heart.

But at the same point, we have had this influx that has taxed all these services.

WILLIAM BRANGHAM: While we were there, we heard that there had been these Facebook rumors of ducks and geese being taken by Haitians out of city parks.

County commissioners said that there were no -- no evidence of that.

But then this pets rumor that was amplified last night really took off when the senator from Ohio, J.D.

Vance, put out this tweet earlier this week.

He said this: "Reports now show that people have had their pets abducted and eaten by people who shouldn't be in this country."

Again, we asked the city manager, and the city manager in a city like Springfield, like in many cities, is like the CEO of the city.

And he again reiterated there are no credible reports that immigrants had taken pets, abducted pets, or harmed pets in any way.

And on some level, there's a logical question you have to ask here.

If you believe that your beloved pet had been stolen and killed and eaten by someone, who would you first contact, the police or your local senator?

The local senator here, J.D.

Vance, argues he's received multiple calls about this.

The local officials say they have received none.

Even today, after the debate, they double down saying all of these things are -- quote -- "rumors" that have been spread by misinformation.

AMNA NAWAZ: William, these very ugly rumors do have some historical context we should get into here, right?

Like, how have these kinds of claims been used against other groups in the past?

WILLIAM BRANGHAM: That is exactly right, Amna.

Researchers who study this field say rumors like this are often used to demonize people who are seen as different, whether you are from another country, speak another language, or simply look different.

We talked earlier, Mary and I, to Cynthia Miller-Idriss.

She's a researcher at American university.

Here's what she had to say.

CYNTHIA MILLER-IDRISS, American University: There's always been this idea of a threat from the other, from immigrants, from people of color to the nation, to white families, to civilization itself, right, this idea that there's some terrible threat coming that has to be defended against, and then, as evidence of that, this idea that your beloved pets are being stolen and eaten, which is patently false.

There's no evidence whatsoever that that's happening.

And it's the kind of thing that makes people scared of the other.

It demonizes and it dehumanizes.

WILLIAM BRANGHAM: So, even though there is no evidence that immigrant communities in America commit violence at greater rates than native-born Americans, the former president and his running mate keep leaning into this argument that we are being invaded by a violent immigrant horde, and it is not true.

AMNA NAWAZ: It's great reporting from you and producer Mary Fecteau.

William Brangham, thank you so much.

WILLIAM BRANGHAM: You're welcome.