Until the morning of April 24, 2013, director Andrew Morgan hadn’t really given much thought to the topic of fashion. But as he grabbed a coffee and a copy of the New York Times that day, he read about the collapse of the Rana Plaza garment-factory building, which killed more than 1,100 people and injured some 3,000 more.

“It definitely broke my heart, but I think the more stunning thing to me when I read that article was the awareness that I had never in my whole life really stopped to think about where my clothes came from,”...