On a recent Friday, Kwanza Brooks, a $7.25-an-hour McDonald’s worker, climbed into a 14-person van to take a four-hour ride from Charlotte, N.C., to Atlanta. As she and other workers headed south, Ms. Brooks, a short, fiery woman, swapped stories with her companions about unsafe conditions and unfair managers. Upon arriving, they joined more than 400 other people — including home care aides, Walmart workers, child care workers and adjunct professors. The gathering was a strategy session to plan for the fast-food movement’s next big wave of protests, which is now scheduled for April 15. But the meeting was also...