WASHINGTON — Vice President Kamala Harris and former President Donald Trump are racing toward swing states that they hope to flip in their favor this year, both of them trying to expand their narrow paths to victory in a closely fought presidential campaign.

Harris has her sights set on North Carolina, where she scheduled rallies in Charlotte and Greensboro on Thursday, her first political events after she buoyed supporters with her performance in Tuesday’s debate. Her team is working to turn key moments from the debate into new television and digital advertising, and promising more travel in battleground states.

Trump headed west to Tucson, Arizona, as he looks to stabilize his campaign, which continues to struggle to recalibrate nearly two months after Harris replaced President Joe Biden at the top of the Democratic ticket.

Although Harris’ team proposed a future debate, Trump on Thursday rejected the idea in a post on his Truth Social account.

“THERE WILL BE NO THIRD DEBATE!” Trump said with his signature all-caps, including his debate against Biden in the count.

The candidates are barnstorming one day after they marked the anniversary of the Sept. 11, 2001, attacks, a somber occasion that provided little respite from partisan politics in a high-speed campaign season.

At a fire station in Shanksville, Pennsylvania, close to where United Airlines Flight 93 crashed after passengers fought back against their hijackers, Trump posed for photos with children who wore campaign shirts. One of the shirts proclaimed former House Speaker Nancy Pelosi, Biden and Harris were “dumb and dumber and dumbest.”

Biden and Harris visited the same fire station earlier in the day. Someone there offered Biden a red, white and blue baseball cap that said “Trump 2024,” and suggested the president put it on to demonstrate his commitment to bipartisan unity. Biden briefly put it on and flashed a wide grin.

Only a handful of battleground states will decide the outcome of the election.

Democrats haven’t won North Carolina’s electoral votes since 2008, when President Barack Obama was elected for the first time. However, Trump’s 2020 margin of victory of 1.3 percentage points was his narrowest win of any state that year, and Democrats hope that North Carolina’s growing and diversifying population will give them an edge this time.

Harris’ campaign said Thursday’s trip was her ninth to the state this year, and recent polls show a tight race. More than two dozen combined campaign offices — supporting Harris and the rest of the party’s candidates — have been opened.

Republicans have expressed confidence in Trump’s chances in the state, and the former president held rallies there in August.

Registered independents — known in North Carolina as unaffiliated — are the state’s largest voting bloc and are usually key to determining outcomes in statewide elections. A state Supreme Court ruling this week affirming that Robert F. Kennedy Jr. must be removed from North Carolina ballots could send additional votes Trump’s way given Kennedy’s endorsement.

The state’s Republican Party has dismissed concerns that a poor showing by its gubernatorial nominee, Lt. Gov. Mark Robinson, could harm the electoral chances of other party candidates, including Trump.

Democratic nominee Josh Stein and his allies have hammered Robinson for months on the airwaves and social media for his past harsh comments on abortion and LGBTQ+ rights. Stein, the state attorney general, had a lead over Robinson in several recent polls of North Carolina voters.

Arizona is another state where the presidential race could be shaped, at least in part, by down-ballot races. Kari Lake, a prominent Republican election denier who lost her campaign for governor in 2020, is running for the U.S. Senate seat that’s being vacated by Sen. Kyrsten Sinema.

She’s opposed by Democratic Rep. Ruben Gallego, who was leading in several recent polls, although the race was close in another.

Republicans have won Arizona in nearly every presidential election since World War II, but Biden eked out a narrow victory in 2020.