Rival markets like rideshare services, universities, and organ exchanges compete to attract participants, seeking to maximize their own utility at potential cost to overall social welfare.  Similarly, individual participants in such multi-market systems also seek to maximize their individual utility. If entry is costly, they should strategically enter only a subset of the available markets. All of this decision making---markets competitively adapting their matching strategies and participants arriving, choosing which market(s) to enter, and departing from the system---occurs dynamically over time. This paper provides the first analysis of equilibrium behavior in dynamic competing matching market systems---first from the points of view of individual participants when market policies are fixed, and then from the points of view of markets when agents are stochastic. When compared to single markets running social-welfare-maximizing matching policies, losses in overall social welfare in competitive systems manifest due to both market fragmentation and the use of non-optimal matching policies. We quantify such losses and provide policy recommendations to help alleviate them in fielded systems.