Mr. Trump believes that focusing on illegal immigration is a winning strategy for him, and his advisers have come to share that view, armed with polling data that show that the issue continues to be a top priority for Republican base voters. They are using the caravan to raise broader fears as well about what they portray as a broken, permissive immigration system that is weakening the country culturally, financially and even physically.Republican officials are aware of the political downside of highlighting the issue, which tends to turn off suburban women and college-educated white voters, but they have calculated that using fear of crime and terrorism can neutralize that disadvantage. And they believe that Democrats have overplayed their hands on immigration because of calls by some of their prominent leaders for the abolition of Immigration and Customs Enforcement, the agency responsible for deportations.That is why Mr. Trump — and many Republican candidates who have followed his lead — have been touting their support for ICE and highlighting MS-13, the brutal transnational gang with roots in El Salvador, as well as the risks of so-called sanctuary cities that limit their cooperation with the federal immigration authorities.Of the dozen people interviewed at Mr. Trump’s rally, almost all of them spoke in considerable detail about their concerns over immigration. Ms. Hooten, the Trump supporter at the rally on Saturday, blamed Mr. Soros, Hillary Clinton and former President Barack Obama for the caravan.“I think they’re all involved in this,” she said. “I feel it’s treason.”Another person at the rally, Jeff Hutson of Columbus, Ky., called the caravan “a smoke screen” — no different from a lot of other big news stories he said he thinks are intended to distract Americans. “I think we’ve had a lot of that going on,” he said. “I think a lot of the shootings we’ve had — anything big to draw attention away from something else that’s going on.”The language and imagery of danger lurking within the caravan started almost immediately after its existence was reported this month. In some cases, as evident from the unproven claim that Middle Easterners were hiding among the migrants, that rhetoric traces a path from right-wing activists online to Mr. Trump.The Center for Immigration Studies, a group the president has cited in the past for its research promoting his immigration policies, was among the first to report on Oct. 16 on claims by the president of Guatemala — made before the caravan had formed — that his country had captured and deported 100 people with ties to ISIS.