The U.S. military said it would send 5,200 troops to the southwest border in response to a caravan of migrants from Central America that President Trump has seized on as an issue in the Nov. 6 midterm elections.

The movement is the largest, quick deployment of active-duty U.S. troops since the response to the 2010 earthquake in Haiti, when thousands arrived within days, former officials said. It is a major escalation of the U.S. response at the border, which as recently as last week was planned for 800 troops. With 2,000...