As they gathered this week in South America, many of the world’s leaders were engaged in a delicate diplomatic dance with President Xi Jinping of China.

Keir Starmer, the British prime minister, called for a “consistent, durable” relationship with China. Anthony Albanese, the prime minister of Australia, pledged to be “patient, calibrated and deliberate.” President Biden promised not to let “competition veer into conflict,” even as he prepared to hand power over to President-elect Donald J. Trump, who has vowed confrontation with China on tariffs.

As the United States makes a transition from Mr. Biden to Mr. Trump, presidents and prime ministers around the world are searching for stability, particularly when it comes to China. Mr. Xi, in his own remarks during a meeting in Peru, told Mr. Biden that he wanted to maintain a “stable, healthy and sustainable” relationship with the United States.

But the steadiness that the world leaders seek with China is threatened by a host of complicated issues that lingered just beneath the veneer of civility at the Group of 20 summit in Rio de Janeiro, which ended on Tuesday.

Potential conflicts with China loom on human rights, the fate of Taiwan, technology competition, cyber attacks, aid to Russia and tariffs.

And for all the pomp and pleasantries as Mr. Biden wraps up a half-century on the world stage, there is deep uncertainty about the role the United States might play in heading off those conflicts.