Demand-Responsive Transit (DRT) operates without fixed routes or timetables. Vehicles are dispatched in response to passengers' real-time requests. DRT is well suited to environments where demand is temporally concentrated, spatially dispersed, and highly variable. These characteristics align with the tourism context, where attractions are spread out and travel demand fluctuates considerably. DRT can serve as an alternative. It enables tourists to flexibly visit multiple dispersed attractions, even in transit-underserved regions with limited public transportation. However, realizing the potential of DRT requires sophisticated operational decisions. Operators must decide which vehicle serves which request, in what sequence, and when. This challenge intensifies because tourist travel requests arrive sequentially throughout operating periods. Operators must periodically revise plans while ensuring service continuity for passengers already onboard. This study develops an optimization model for DRT operations under such dynamic conditions. The problem is formulated as a mixed-integer linear programming (MILP) model. It is based on the standard structure of the vehicle routing problem (VRP) and the dial-a-ride problem (DARP) and is extended to account for mid-operation re-planning. The model fixes the assigned vehicle for onboard requests to guarantee service continuity. Vehicles resume operation from their current locations rather than the depot and may depart early. This static model is applied repeatedly via a rolling-horizon procedure. The system updates vehicle states and request sets at fixed intervals. It responds to dynamically arriving demand while preserving solution quality at each stage. The model is implemented using IBM ILOG CPLEX and validated through a small-scale numerical example. The results confirm that the re-optimization procedure maintains the continuity of onboard passengers while efficiently integrating newly arriving requests, with idle vehicles dispatched only when additional capacity is required. This study thus provides a foundation for real-time DRT operational decision-making that supports access to dispersed attractions in transit-underserved tourism settings.