The 44th president sharply criticized Mr. Trump, who hit back later in the day. Mr. Obama also assailed Vivek Ramaswamy, the Republican nominee for governor of Ohio.

© Victor J. Blue for The New York Times

Supporters of Mike Rogers, the Republican candidate, have described his Democratic opponent, Dr. Abdul El-Sayed, as un-American and worse.

© Emily Elconin for The New York Times; Chip Somodevilla/Getty Images

The transfer came a day after the news that the pro-Israel lobby was planning to contribute as much as $9 million to defeat the Democratic nominee for Senate.

© Sarah Rice/Getty Images

Abdul El-Sayed accused Mike Rogers of being overly devoted to President Trump. Mr. Rogers accused Dr. El-Sayed of not enough loyalty to the country. It was still more cordial than their last debate.

© Pool photo by Robin Buckson