In many dynamic matching markets, priorities depend on previous allocations. In such environments, agents on the proposing side can manipulate the period-by-period deferred acceptance (DA) mechanism. We show that the fraction of agents with incentives to manipulate the DA mechanism approaches zero as the market size increases. In addition, we provide a novel algorithm to calculate the percentage of markets that can be manipulated. Based on randomly generated data, we find that the DA becomes approximately non-manipulable when the schools capacity reaches 20. Our theoretical and simulation results together justify the implementation of the period-by-period DA mechanism in dynamic markets.