Described are medical products, systems, and methods useful for treating fistulae, particularly enterocutaneous fistulae. Certain inventive products are configured to have portions residing in and around a primary fistula opening in a wall of the alimentary canal. One product includes a graft body which is configured to block at least the primary opening. The graft body includes a capping member connected to an elongate plug member. The capping member is configured to contact portions of the alimentary canal wall adjacent to the primary opening, with the elongate plug member extending into at least a portion of the fistula. A product of this sort may be particularly adapted to allow a portion of the capping member to be positioned alongside an exterior, lateral surface of the plug member, e.g., when placed in a delivery device lumen. Such a capping member may be hingedly or non-hingedly coupled to the elongate plug member.
