  In this article, relying on Foster-Lyapunov drift conditions, we establish
subexponential upper and lower bounds on the rate of convergence in the
$\mathrm{L}^p$-Wasserstein distance for a class of irreducible and aperiodic
Markov processes. We further discuss these results in the context of Markov
L\'evy-type processes. In the lack of irreducibility and/or aperiodicity
properties, we obtain exponential ergodicity in the $\mathrm{L}^p$-Wasserstein
distance for a class of It\^{o} processes under an asymptotic flatness (uniform
dissipativity) assumption. Lastly, applications of these results to specific
processes are presented, including Langevin tempered diffusion processes,
piecewise Ornstein-Uhlenbeck processes with jumps under constant and stationary
Markov controls, and backward recurrence time chains, for which we provide a
sharp characterization of the rate of convergence via matching upper and lower
bounds.
