We discuss existence and uniqueness of stationary and ergodic nonlinear autoregressive processes when exogenous regressors are incorporated in the dynamic. To this end, we consider the convergence of the backward iterations of dependent random maps. In particular, we give a new result when the classical condition of contraction on average is replaced with a contraction in conditional expectation. Under some conditions, we also derive an explicit control of the functional dependence of Wu (2005) which guarantees a wide range of statistical applications. Our results are illustrated with CHARN models, GARCH processes, count time series, binary choice models and categorical time series for which we provide many extensions of existing results.