•We model a two-party dynamic voting model with heterogeneous voters.•Voters’ preferences endogenously evolve over time and parties are policy motivated.•Model’s outcomes include convergence, instability, and extreme equilibria.•Convergence is achieved when neither party has enough support to pursue its agenda. We test whether the original median voter theorem’s result of political convergence in a two-party system holds when the control variables that influence voters’ preferences endogenously evolve over time and parties are policy motivated. We present a dynamic voting model in which voters’ heterogeneous preferences evolve over time depending on observable common factors and unobservable idiosyncratic characteristics. In such a setting, the convergence of parties’ platforms to the centre is a special case within a range of results that include instability and extreme equilibria. In particular, convergence is achieved not as the result of electoral strategies, but when neither party has enough support to pursue its agenda.