Hours after the presidential debate, Vladimir Putin’s spokesman, Dmitri Peskov, said that his boss didn’t bother to watch. Mr. Peskov even scoffed at the notion: The Russian president could hardly be expected to set his alarm to watch the broadcast in the middle of the night in Moscow. In any case, he added, the debate was America’s internal affair. But judging by the wave of propaganda that immediately followed on Russian television, Mr. Putin was, indeed, paying attention.

For Russia’s leader, like his Chinese counterpart, Xi Jinping, the crisis in American leadership that crystallized for the world to see onstage last week was a gift — the perfect opportunity for their governments to re-up the narrative of America’s imminent collapse they’ve been pushing. By engendering uncertainty about American global leadership, Mr. Putin and Mr. Xi believe they can reorder the world in a way that amplifies their countries’ international influence at Washington’s expense and helps preserve their grip on power.

Russia’s state-owned Channel One covered the debate in depth, describing President Biden and Donald Trump as “small children from the nursery.” As Russian viewers were treated to replays of the candidates bickering over who was the better golfer, the presenter predicted that life itself in America would become “one never-ending game of golf,” with the United States lurching “from one big hole to the next.”

China’s official commentary was more reserved. The state media briefly summarized the debate, also focusing on the barbs and the insults. Beijing’s Global Times mouthpiece quoted an anonymous American voter saying, “There must be something wrong with the system,” and another claiming that the U.S. democracy was failing.

This is all part of the long game for Mr. Putin and Mr. Xi, who for years have sought to counter accusations that their regimes are tyrannies by attacking American democracy. They have tried to plant the idea among their own people and their sympathizers in the West and developing nations that the United States is not a democracy at all but a dysfunctional oligarchy in terminal decline.

In February 2022, a few weeks before Russia’s invasion of Ukraine, Mr. Putin and Mr. Xi signed a statement claiming that Russia and China had “deep democratic traditions, based on a thousand years of development,” and criticized certain unnamed countries (read: the United States) for attempting to impose their own version of the political system on others in a way that “undermines democracy and discards its spirit and true values.”