Rideshare platforms such as Uber and Lyft dynamically dispatch drivers to match riders' requests. We model the dispatching process in rideshare as a Markov chain that takes into account the geographic mobility of both drivers and riders over time. Prior work explores dispatch policies in the limit of such Markov chains; we characterize when this limit assumption is valid, under a variety of natural dispatch policies. We give explicit bounds on convergence in general, and exact (including constants) convergence rates for special cases. Then, on simulated and real transit data, we show that our bounds characterize convergence rates -- even when the necessary theoretical assumptions are relaxed. Additionally these policies compare well against a standard reinforcement learning algorithm which optimizes for profit without any convergence properties.