  This paper brings some insights of $\psi'$-mixing, $\psi^*$-mixing and
$\psi$-mixing for copula-based Markov chains and the perturbations of their
copulas. We provide new tools to check Markov chains for $\psi$-mixing or
$\psi'$-mixing, and also show that perturbations of $\psi'$-mixing copula-based
Markov chains are $\psi'$-mixing while perturbations of $\psi$-mixing Markov
chains are not necessarily $\psi$-mixing Markov chains, even when the perturbed
copula is $\psi-mixing$. Some examples of copula families are considered. A
statistical study is provided to emphasize the impact of perturbations on
copula-based Markov chains. Moreover, we provide a correction to a statement
made in Longla and al. (2021) on $\psi$-mixing.
