There is little evidence that Mr. Trump’s recipe for social media dominance translates well to smaller races. But many candidates are trying it anyway.

Jim Hagedorn, a Republican running in Minnesota’s First Congressional District, has attacked his Democratic opponent, Dan Feehan, for supporting the former N.F.L. quarterback Colin Kaepernick’s protest of police brutality during the national anthem, calling him “part of the Kaepernick wing of the Democrat Party.” Scott Hawkins, a Republican who ultimately withdrew from this year’s Alaska governor’s race, took out a Facebook ad during his campaign promoting his opposition to MS-13, a street gang that Mr. Trump has vocally opposed, saying “I’m with The Donald on this one.” (MS-13, which originated in Los Angeles, is not known to have a presence in Alaska.)

And the Supreme Court nomination of Justice Brett M. Kavanaugh lit up down-ballot races across the country, as candidates running for judgeships and district attorneys’ offices raced to signal their full-throated support for either him or his opponents. Several of the highest-performing posts on Facebook during then-Judge Kavanaugh’s confirmation fight came not from national news outlets like CNN or Politico but from Sid Miller, the Texas agriculture commissioner, who is running for re-election in November. In one post, which received nearly 20,000 shares, he praised Judge Kavanaugh’s wife, Ashley, and urged people to pray for her.

“All the oxygen is in these broad, national debates,” said Rob Flaherty, the creative director of Priorities USA, a progressive organization that is among the biggest liberal buyers of digital advertising. “If you want to do the Trump tactic of stealing attention, you have to talk about those things.”

Aiming for mainstream appeal can also be a savvy financial move. Facebook’s advertising system is built on an auction process that takes into account the likelihood that a given ad will provoke an audience to engage with it. More engaging ads can cost less to buy than similar ads with less engaging language, and they can spread further as users share them with their friends.

“Trump was the first candidate to run an internet-first campaign,” said Gerrit Lansing, a former chief digital officer of the Republican National Committee. “However, the vast majority of campaigns and candidates are still consulted by people who spent their careers with newspapers and the evening news, and that’s just not how the world works anymore. Your political branding needs to be punchier in the internet age.”

Ben Kalasho, a city councilman in El Cajon, Calif., outside San Diego, may be taking Mr. Trump’s act to its logical extreme. Mr. Kalasho, 35, is a flashy businessman who oversees a local beauty pageant, lives in a mansion with his wife, a former model, and has been mired in legal trouble, including allegations of fraud and sexual harassment. He has denied those accusations.