Could anyone give me some references for the convergence rate of Markov chain arising from the random iteration of Lipschitz functions which moves as follows:

$$ X_{n+1}= f_{\omega_n}(X_n)$$ where $f_1,\dots, f_s$ are Lipschitz functions (with Lipschitz constants $L_i$ and $\sum_{k=1}^{s} p_kL_k <1$ )on $\mathcal X$, a metric space with a metric $d$ it is complete and separable say. $\omega_0,\omega_1,\dots, \omega_n,\dots$ are i.i.d discrete random variable taking values in $\{1,2,\dots,s<\infty\}$. $p_k=\mathbb P(\omega_i=k)$.

I found Diaconis and Freedman's Paper where they didn't talk about the finite number of function though and convergence rate is exponential in Prohorov's metric. I am looking for some reference on Wasserstein metric or so. Thanks for helping!