We establish existence and continuity properties in a model of dynamic elections with a discrete (countable) state space and general policies and preferences. We provide conditions under which there is a representative voter in each state, and we give characterization results in terms of the equilibria of an associated “representative voting game.” When the conditions for these results are not met, we provide examples that uncover new classes of dynamic political failures. The tractability of the model is illustrated in several applications, including a model of public investment with two economic states, a model of ideological politics with two demographic states, and a model of changing politician valence that captures either experience or corruption.