For the second time in a little over a month, Senator Mitch McConnell of Kentucky, the longtime Republican leader, froze up during a news conference on Wednesday, elevating concerns about his health and his ability to complete his term that ends in January 2027.

At an event hosted by the Northern Kentucky Chamber of Commerce, Mr. McConnell, 81, who was elected to his seventh term in 2020, paused for about 30 seconds while responding to a reporter’s question about his re-election plans.

The abrupt spell — like one at the U.S. Capitol in July — happened in front of the cameras. In March, a fall left him with a concussion. He suffered at least two other falls that were not disclosed by his office.

Mr. McConnell has brushed off past questions about his health, but speculation is swirling again about what would happen in the unlikely event that he retired in the middle of his term.

How would the vacancy be filled?

For decades in Kentucky, the power to fill a vacancy in the U.S. Senate was reserved exclusively for the governor, regardless of whether an incumbent stepped down, died in office or was expelled from Congress.

But with Gov. Andy Beshear, a Democrat, in the state’s highest office, Republican lawmakers used their legislative supermajorities to change the state law in 2021.

Under the new law, a state executive committee consisting of members of the same political party as the departing incumbent senator will name three candidates the governor can choose from to fill the vacancy on a temporary basis. Then a special election would be set, and its timing would depend on when the vacancy occurs.

At the time that G.O.P. lawmakers introduced the change, Mr. McConnell supported the measure. Mr. Beshear, who is up for re-election this November, vetoed the bill, but was overridden by the Legislature.

Who might follow McConnell in the Senate?

In a state won handily by former President Donald J. Trump, several Republicans could be in the mix should Mr. McConnell, the longest-serving leader in the Senate, step down.

But replacing him with a unflagging ally of the former president could rankle Mr. McConnell, who has become a fairly sharp, if cautious, critic of Mr. Trump after the former president’s attempts to overturn the 2020 election and after the Jan. 6, 2021, attack on the U.S. Capitol.

One name to watch could be Daniel Cameron, the state’s attorney general, who is challenging Mr. Beshear in the governor’s race and has been considered at times an heir apparent to Mr. McConnell.

Should he lose his bid for governor — which drew an early endorsement from Mr. Trump — talk of succession could be inevitable despite his connection to the former president.

Ryan Quarles, the well-liked agricultural commissioner, might also be a contender. He lost this year’s primary to Mr. Cameron in the governor’s race.

Kelly Craft, a former U.N. ambassador under Mr. Trump, who finished third in that primary, has the political connections to seemingly be part of the conversation. She is married to a coal-industry billionaire, who spent millions on advertising for her primary campaign.

And then there is Representative Andy Barr, who has drawn comparisons to Mr. McConnell and who described Mr. Trump’s conduct as “regrettable and irresponsible,” but voted against impeachment after the riot at the Capitol.

What have McConnell and his aides said about his health?

Both times that Mr. McConnell froze up in front of the cameras, his aides have said that he felt lightheaded.

But his office has shared few details about what caused the episodes or about his overall health. He missed several weeks from the Senate this year while recovering from the concussion in March, which required his hospitalization.

Mr. McConnell, who had polio as a child, has repeatedly played down concerns about his health and at-times frail appearance.

“I’m not going anywhere,” he told reporters earlier this year.

How is Congress dealing with other lawmakers’ health issues?

For the current Congress, the average age in the Senate is 64 years, the second oldest in history, according to the Congressional Research Service.

Senator Dianne Feinstein, a Democrat from California who is the chamber’s oldest member at 90, has faced health problems this year that have prompted growing calls for her to step down.

In February, she was hospitalized with a severe case of shingles, causing encephalitis and other complications that were not publicly disclosed. She did not return to the Senate until May, when she appeared frailer than ever and disoriented.

This month, she was hospitalized after a fall in her San Francisco home.

Longtime senators are not the only ones in the chamber grappling with health concerns.

John Fetterman, a Democrat who was Pennsylvania’s lieutenant governor, suffered a near-fatal stroke last May and went on to win one of the most competitive Senate seats in November’s midterm elections.

Nick Corasaniti and Annie Karni contributed reporting.