We establish a general framework to study the rate of convergence of a Euler type approximation scheme with decreasing time steps to the invariant measure, for a general class of stochastic systems. The error is measured in general Wasserstein distances, which enables to encompass cases with non global contractivity conditions. Our main assumption is a coupling property which is expressed in terms of the one-step approximation. We show that the proposed set-up can be applied to a wide range of equations that may be law dependent, such as Langevin equations, reflected equations, Boltzmann type equations and for a recent McKean Vlasov type model for neuronal activity.