Demand responsive transit (DRT) has the potential to provide passengers with higher accessibility and lower travel time as compared with conventional transit, and at the same time make more efficient use of vehicle capacity than traditional taxi. In many current systems, vehicles are assigned to passengers along travel paths that are chosen myopically. When information on future demand distribution is available, it would be more beneficial to dispatch transit vehicles strategically to areas with a higher probability of generating passengers. This paper proposes a mathematical model for a dynamic DRT vehicle dispatch problem. It determines in real time how the operating vehicles shall be used to serve arriving passenger demand, and which paths the vehicles should choose to achieve a balance between operator and passenger costs. The model is solved by an approximate dynamic programming (ADP) based solution approach. Case studies, including a hypothetical numerical example and a real-world case in Qingdao, China, have been conducted to demonstrate the applicability of the proposed modeling framework. Results show that the proposed ADP solution can significantly improve the overall system performance as compared with myopic benchmarks.