In analyzing complex systems modeled by stochastic partial differential equations (SPDEs), such as certain turbulent fluid flows, an important question concerns their long-time behavior. In particular, one is typically interested in determining how long it takes for the system to settle into statistical equilibrium, and in investigating efficient numerical schemes for approximating such long-time statistics. In this talk, I will present two general results in this direction, and illustrate them with applications to the 2D stochastic Navier-Stokes equations. Specifically, our results provide a general set of conditions that guarantee: (i) Wasserstein contraction for a given Markov semigroup and, consequently, exponential mixing rates; and (ii) uniform-in-time weak convergence for a parametrized family of Markov semigroups towards a limiting dynamic. Most importantly, our approach does not require gradient bounds for the underlying Markov semigroup as in previous works, and thus provides a flexible formulation for a broad range of applications. This is based on joint work with Nathan Glatt-Holtz (Tulane U).