“I relate to him more, and he has legislative experience,” said Mr. Burke, 33, who wore a gingham shirt and blue blazer and joked that he was one of downtown’s “few young Republicans.”

Registered Democrats living in Washington outnumber Republicans more than 10 to one, but Patrick Mara, the executive director of the DC Republican Party, noted the national party had granted it 19 delegates to the convention, not an insignificant number considering that all of Florida, the biggest state voting on Tuesday and many times the size of the capital, has 99.

As a result, Mr. Mara said he was surprised candidates did not campaign here, though he acknowledged, “It’s hard to run here in a public way when you are spending your whole campaign running away from Washington.”

Wyoming represented the day’s other prize. Three of the state’s 29 delegates are unpledged state party officials, and only 12 delegates were contested on Saturday, with Mr. Cruz, the Texas senator, winning nine of them. The remaining 14 will be pledged at a state convention on April 16. Officials in Wyoming have begun studying whether to abandon their complicated voting system, which involves three separate elections, and move to a primary.

“We don’t see a lot of attention,” explained Tom Wiblemo, executive director of the Wyoming Republican Party.

But the Wyoming party’s chairman, Matt Micheli, pointed out that Mr. Cruz had visited in August, hosting a couple of large rallies on opposite ends of the state, and that the Cruz campaign had remained engaged throughout the primary season. Donald J. Trump never made it to the state, Mr. Kasich visited last year and Rubio surrogates held several events.

Saturday’s elections actually began on Friday evening, Eastern Time, in Guam, about 8,000 miles from Washington and on the other side of the international date line. About 300 Republicans met in a hotel ballroom there to vote to send nine delegates to the party’s convention in Cleveland.