Demand-responsive transport (DRT) services (collective on-demand services, such as shared ridesourcing and microtransit) offer a collective flexible travel alternative that can potentially complement fixed transit (FT). The combination of an on-demand and line-based service holds the promise of improved mobility and increased service coverage. However, to date, it remains unknown whether DRT services deliver these much anticipated improvements. This study presents an assessment framework to evaluate the performance of DRT and related changes in accessibility, and performs an empirical analysis for a recently introduced DRT service in the Netherlands. The framework includes a performance benchmark between DRT and FT based on the computation of generalized journey times of the DRT rides and the FT alternatives, and can help identify whether DRT is used as a complement or a substitute for FT. The framework covers the spatial and temporal dimensions, and the explicit consideration of rejected trips is an integral part of the evaluation. Results suggest large accessibility improvements for DRT users, especially for some underserved origin–destination pairs.