In this work we investigate the long-time behavior for Markov processes obtained as the unique mild solution to stochastic partial differential equations in a Hilbert space. We analyze the existence and characterization of invariant measures as well as convergence of transition probabilities. While in the existing literature typically uniqueness of invariant measures is studied, we focus on the case where the uniqueness of invariant measures fails to hold. Namely, introducing a generalized dissipativity condition combined with a decomposition of the Hilbert space, we prove the existence of multiple limiting distributions in dependence of the initial state of the process and study the convergence of transition probabilities in the Wasserstein 2-distance. Finally, we apply our results to Levy driven Ornstein–Uhlenbeck processes, the Heath–Jarrow–Morton–Musiela equation as well as to stochastic partial differential equations with delay.