A DTL is basically responsible for everything your STL is responsible for but for the entire district. VIBE, AAR, Conversion, Payroll, etc but on a district level. They are leading their team (STL direct reports) to drive sales. They are also active with their peers in the group driving results across all stores in said group.
Our DTL actually sent out this pompous, contradicting newsletter. At the beginning of the letter, it talked about how great we all are. Halfway down the letter, it bashed all the district stores. It reminded me of the yearly employee reviews: "You do great in this area, but in this same exact area you do shit. Do better, or your ass is grass!"