‘It Is Evening, Isn’t It?’ An 80-Year-Old President’s Whirlwind Trip.

On the 13th hour of the third day of a two-country trip, President Biden stepped onstage at a news conference in Vietnam and bid reporters there a good evening.

At least, he thought so.

“It is evening, isn’t it?” the president said, drawing laughs from the jet-lagged masses. “This around-the-world-in-five-days is interesting, isn’t it?”

He was joking, but only sort of.

The trip, which began in New Delhi with the Group of 20 summit, was a whirlwind for Mr. Biden. He went abroad with a list of diplomatic to-dos, most of which were aimed at signaling to China that the United States was working to line up allies who are fed up with Beijing’s aggression in the region. In Hanoi, he celebrated the elevation of the U.S.-Vietnamese partnership to the highest level in Vietnam’s diplomatic hierarchy, and said it was part of his administration’s strategy to bolster the American presence in the Indo-Pacific.

But Mr. Biden took another objective overseas, too, as he enters an election season facing questions about his age and stamina: showing that he is still up to the challenges of globe-trotting statesmanship.

His aides described a president in near constant motion. Before traveling to Hanoi, advisers said that Mr. Biden had met with more than 30 world leaders, mostly in informal meet-and-greets, at the G20 summit in India. The interactions, they said, were designed to be more ad hoc than the traditional bilateral meetings that accompany international summits.

Before Saturday’s diplomacy, a priest went to Mr. Biden’s hotel for Mass, an arrangement the president maintains anytime one of his foreign trips happens over a weekend, according to aides who asked not to be named because they were not authorized to publicly discuss behind-the-scenes movements.

The president also told aides that he wanted to reach out to the tennis player Coco Gauff, who had just won the U.S. Open, so they worked to find time for a call. (Aides said he’d been following the tournament.)

With all the running around, it seemed like little coincidence when, in Hanoi, Mr. Biden, 80, repeated himself: “These five-day trips around the world are no problem,” he told a BBC reporter who confirmed the time of day.

Mr. Biden’s trip may ultimately do little to alter his political fortunes back home, where his polling numbers are low and he seems headed toward a tight race against his predecessor, Donald J. Trump. House Republicans are itching to push ahead on a long-shot impeachment inquiry, and his son, Hunter, is facing federal indictment on a gun possession charge.

The president is also facing persistent questions about his age and effectiveness as a messenger for his own agenda — part of a broader conversation in Washington about leaders, some with health problems, who remain in office well into their 80s or 90s.

As aging politicians of both parties, including Senator Mitch McConnell of Kentucky and Senator Dianne Feinstein of California, suffer from health complications in the public eye, some are beginning to call for more transparency into the medical history of elected leaders.

“I think if you want to be the president of the United States, or a senator or House member, then there is a responsibility over and above that of just offering yourself,” Bill Cassidy, a Republican senator from Louisiana and a physician, said in an interview on “Meet the Press.” “It has to be that you can show that you have clarity.”

Back home and abroad, White House officials have publicly expressed irritation over news reports that describe the president as keeping the same languid campaign schedule as Mr. Trump, who is spending most of his time filming videos railing against his indictments and collecting endorsements from other Republicans. Several pointed to Mr. Biden’s Asia trip as evidence that he was busy with governing.

In the past, aides have adjusted schedules to accommodate the daily rhythms of the oldest president in American history, including last year, when initial plans for a 10-day trip to the Middle East and Europe were split into separate journeys. On more recent trips abroad, Mr. Biden had skipped evening dinners, absences that aides attributed to the president’s desire to prepare for the next day or check in on things back home.

This time, in India, Mr. Biden made the dinner. And on Monday in Hanoi, he was up early for a day of activities that included a visit to a memorial dedicated to the military service of John McCain and to attend a luncheon — where officials toasted to his health — before traveling back to the United States. On the 30-hour leg home, Mr. Biden was scheduled to address troops in Alaska to commemorate the anniversary of the Sept. 11 attacks.

Ben LaBolt, the White House communications director, who was on the trip with the president, said that the schedule was not intended to make a point. “President Biden’s focus on strengthening alliances and building relationships with world leaders is a longtime hallmark of his career.” (The president was chairman of the Senate Foreign Relations Committee for part of his 36 years in the chamber.)

Even if Mr. Biden’s age wasn’t a topic of conversation among his aides, it was apparently on the mind of at least one other leader. Nguyen Phu Trong, the 79-year-old general secretary of the Communist Party of Vietnam, made a point of publicly commenting about Mr. Biden’s vitality.

“You have nary aged a day and I would say you look even better than before,” said Mr. Trong, who saw Mr. Biden on a trip to Washington in 2015. “Every feature of you, Mr. President, is very much complimentary of your image.”

The president laughed appreciatively.

At the news conference in Hanoi, Mr. Biden grabbed a microphone and walked around the stage answering questions about China, Russia and Ukraine. Not all of his responses were direct, however. At one point, he invoked John Wayne in a long soliloquy about the evolving global view on climate change.

“And there’s a — my — my brother loves havin’ — there’s famous lines from movies that he always quotes. You know, it — and — one of the — is — the — there’s a movie about John Wayne,” the president began, continuing in stops and starts until he finally made his point about global warming.

Later, he seemed unsure of which reporter to call on from a preselected list furnished by aides — “No, I ain’t calling on you,” he told reporters who raised their hands, hoping to fill the void.

Near the end of his 25-minute appearance, Mr. Biden told reporters he was ready to go to bed.

But then he continued taking questions, including one from a reporter who asked him if he’d met with the Chinese premier, Li Qiang, who attended the G20 summit after China’s top leader, Xi Jinping, decided to skip it.

“We talked about what we talked about at the conference overall. We talked about stability,” Mr. Biden said. “It wasn’t confrontational at all.”

A reporter then asked if he was putting the United States’ strategic interests over human rights issues in Vietnam. As Mr. Biden answered — “I’ve raised it with every person I met with,” he said — the voice of his press secretary came over the loudspeaker, the public-relations equivalent of a shepherd’s hook, telling the world the event was over.

Music began playing, and the president left the stage.

It was not yet bedtime.

Aides said that Mr. Biden finished the day by posing for photographs with U.S. Embassy staff members and military officials before receiving a briefing from Secretary of State Antony J. Blinken on the deadly earthquake in Morocco.