BEIJING — Prayer breakfast speeches in Washington are rarely, if ever, the source of bruised feelings overseas. Not so this year.

President Obama’s comments at this week’s religious gathering not only caused domestic problems — offending some conservative American Christians, who objected to the reference to “terrible deeds” during the Crusades — but have also left China and India piqued. The Chinese objected to the warmth shown to the Dalai Lama, while an Indian official reacted to the president’s remarks about acts of religious intolerance in the country by noting that India (read: not the United States) had openly supported the Dalai Lama for decades.

For both India and China, Mr. Obama’s remarks Thursday may have come as something of a surprise.

Just over a week ago, Mr. Obama completed a three-day visit to India aimed at reinvigorating a frayed relationship. And until this week, China and the United States had worked out a reliable pas de deux over the Dalai Lama, 79, the Nobel laureate and Tibetan spiritual leader whom Beijing sometimes describes as a “wolf in sheep’s clothing.”

It goes like this: Chinese leaders warn the White House against granting the Dalai Lama a public audience, and the American president either ignores the threats of diplomatic fallout or finds a way to meet that will result in the least offense to Beijing.