AMNA NAWAZ: With little more than a month to go, the Republican and Democratic presidential campaigns are dialing in on key swing states.

GEOFF BENNETT: And for the first and only time, vice presidential candidate Senator J.D.

Vance and Governor Tim Walz squared off on the debate stage.

Lisa Desjardins has this report.

LISA DESJARDINS: In Georgia today, vice President Kamala Harris arrived to survey the aftermath of Hurricane Helene.

KAMALA HARRIS, Vice President of the United States (D) and U.S. Presidential Candidate: I'm here today to thank all of those who are working to get folks the support and the relief that they so desperately need and so rightly deserve, and particularly devastating in terms of the loss of life that this community has experienced, the loss of normalcy and the loss of critical resources.

LISA DESJARDINS: Meanwhile, fresh off last night's CBS vice presidential debate, Minnesota Governor Tim Walz kicked off a Pennsylvania bus tour.

GOV.

TIM WALZ (D-MN), Vice Presidential Candidate: From her first day as a prosecutor to right now as vice president, Kamala Harris has only one client, the people, the people, the people.

(CHEERING) (APPLAUSE) LISA DESJARDINS: And Donald Trump's running mate, Ohio Senator J.D.

Vance, rallied crowds in Michigan.

SEN. J.D.

VANCE (R-OH), Vice Presidential Candidate: We already ran this experiment once.

Donald Trump's economic policies worked for American families.

They worked for American consumers.

NORAH O'DONNELL, Moderator: Governor, Senator, thank you for joining us.

LISA DESJARDINS: In the vice presidential debate last night, Vance came out focused on Harris.

SEN. J.D.

VANCE: Governor Walz, you blame Donald Trump.

Who has been the vice president for the last 3.5 years?

And the answer is your running mate, not mine.

LISA DESJARDINS: While Governor Walz had a shake your start.

He was asked about the Middle East and whether Israel should preemptively strike at Iran.

GOV.

TIM WALZ: Iran -- or Israel's ability to be able to defend itself is absolutely fundamental, getting its hostages back, fundamental, and ending the humanitarian crisis in Gaza.

LISA DESJARDINS: At the top of Senator Vance's agenda, immigration.

Moderators asked him about former President Trump's vow to carry out mass deportations.

SEN. J.D.

VANCE: I think the first thing that we do is, we start with the criminal migrants.

About a million of those people have committed some form of crime, in addition to crossing the border illegally.

LISA DESJARDINS: The contrast was clear, but the tone civil.

Walz blamed Trump for blocking a relatively conservative border bill and criticized Vance for inflammatory false statements about Haitian migrants in Springfield, Ohio.

GOV.

TIM WALZ: I believe Senator Vance wants to solve this.

But by standing with Donald Trump and not working together to find a solution, it becomes a talking point.

And when it becomes a talking point like this, we dehumanize and villainize other human beings.

SEN. J.D.

VANCE: The people that I'm most worried about in Springfield, Ohio, are the American citizens who have had their lives destroyed by Kamala Harris' open border.

LISA DESJARDINS: The moderators stepped in with one of just two fact-checks in the debate, sparking a fiery exchange.

MARGARET BRENNAN, Moderator: And just to clarify for our viewers, Springfield, Ohio does have a large number of Haitian migrants who have legal status, temporary protected status -- Norah.

NORAH O'DONNELL: Thank you.

SEN. J.D.

VANCE: Well, Margaret, but no.

(CROSSTALK) MARGARET BRENNAN: Senator, we have so much to get to.

Thank you -- Norah.

NORAH O'DONNELL: We're going to turn now to the economy.

Thank you.

SENATOR J.D.

VANCE: Margaret, I think it's important, because the debate -- Margaret, the -- the rules were that guys weren't going to fact-check.

LISA DESJARDINS: Another substantive contrast came over abortion.

Vance was asked about his previous support of a national 15-week ban.

He called it a national standard and pivoted to say Republicans have to do better in explaining their position.

SEN. J.D.

VANCE: As a Republican who proudly wants to protect innocent life in this country, who proudly wants to protect the vulnerable, is that my party, we have got to do so much better of a job at earning the American people's trust back on this issue where they, frankly, just don't trust us.

And I think that's one of the things that Donald Trump and I are endeavoring to do.

I want us as a Republican Party to be pro-family in the fullest sense of the word.

LISA DESJARDINS: But Walz found his footing and blasted Vance's words as masking a GOP that he sees as oppressing women's rights.

GOV.

TIM WALZ: How can we as a nation say that your life and your rights as basic as the right to control your own body is determined on geography?

Donald Trump is trying to figure out how to get the political right of this.

I agree with a lot of what Senator Vance said about what's happening.

His running mate, though, does not, and that's the problem.

LISA DESJARDINS: The debate was often substantive and notably more congenial than past face-offs, ending on a question and key exchange about democracy.

Vance in the past has said he would have tried to block the 2020 results.

He was asked if he would challenge election results this year if he loses.

SEN. J.D.

VANCE: Look, what President Trump has said is that there were problems in 2020, and my own belief is that we should fight about those issues, debate those issues peacefully in the public square, and that's all I have said and that's all that Donald Trump has said.

LISA DESJARDINS: But Walz criticized Vance for not acknowledging the election results and Trump's relationship to the January 6 Capitol riot.

GOV.

TIM WALZ: This is one that we are miles apart on.

This was a threat to our democracy in a way that we had not seen, and it manifested itself because of Donald Trump's inability to say -- he is still saying he didn't lose the election.

I would just ask that.

Did he lose the 2020 election?

SEN. J.D.

VANCE: Tim, I'm focused on the future.

Did Kamala Harris censor Americans from speaking their mind in the wake of the 2020 COVID situation?

(CROSSTALK) GOV.

TIM WALZ: That is a damning nonanswer.

LISA DESJARDINS: Elsewhere, at a Wisconsin news conference last night, Trump himself also had a telling nonanswer to the same question.

QUESTION: Do you trust the process this time around?

DONALD TRUMP, Former President of the United States (R) and Current U.S. Presidential Candidate: I will let you know in about 33 days.

LISA DESJARDINS: Closing in on one month left and still many questions about how this ends.

For the "PBS News Hour," I'm Lisa Desjardins.