Stability analysis consists of identifying conditions under which the number of jobs in a system is guaranteed to remain bounded over time. To date, such long-run performance guarantees have not been available for periodic approaches to dynamic scheduling problems. However, stability has been extensively studied in queueing theory. In this paper, we introduce stability to the dynamic scheduling literature and demonstrate that stability guarantees can be obtained for methods that build the schedule for a dynamic problem by periodically solving static deterministic sub-problems. Specifically, we analyze the stability of two dynamic environments: a two-machine flow shop, which has received significant attention in scheduling research, and a polling system with a flow-shop server, an extension of systems typically considered in queueing. We demonstrate that, among stable policies, methods based on periodic optimization of static schedules may achieve better mean flow times than traditional queueing approaches.