To a reporter who asked him, in Spanish, if it had been his idea to deliver a speech in Spanish, Mr. Kaine replied, in Spanish, that yes, it had, and that it made sense to do it in Arizona because of the demographic makeup of the state.

If all projections hold, Latinos will be the majority by 2030. And this may well be the year when enough Latino voters turn out to deliver Arizona to a Democrat, which would be the first time since another Clinton, Bill, won the state in 1996.

“Vamos a votar,” Mr. Kaine said. Let’s vote.

The community center in Maryvale, between a public library and a public pool, ordinarily hosts jazz, ballet and belly dancing classes for children and adults; private music instructions; and sports leagues. (Two basketball hoops were folded up above the stage.) On Thursday, it had a mix of elected officials, community activists, immigration advocates, union leaders and teachers wearing T-shirts that read “Educators for Hillary” and “Families Fighting Back.”

Everyone who spoke at the rally did so in Spanish, from Ray Martinez, a candidate for the State House of Representatives who read entirely from a script, to Thomas E. Perez, the United States secretary of labor, who spoke off the cuff — mixing up pronouns and genders, but getting his message across. There were plenty of English-only speakers in the crowd, including people of Hispanic heritage, look and last names like Salgado and Benevidez.

One of them was Lorenzo Sierra, a city councilman from Avondale, home to Arizona’s second-largest Latino community. He said his parents forbade him from speaking Spanish at home some 40 years ago, fearing that if he did so, he would not be able to learn English. It is a common narrative among older generations of Hispanics in the United States.

Mr. Sierra did not mind that he could not quite understand what Mr. Kaine was saying. That he spoke Spanish at all, and for a whole 30 minutes, “speaks volumes of the importance of our community,” Mr. Sierra said. “It’s a sign of respect that we haven’t seen from anyone else on this and other presidential campaigns.”