Let $K(\cdot|\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A|\omega)$ for each (Borel) measurable $A$. 

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \epsilon$ then $d(\mu K,\nu K)\leq \epsilon\cdot C$, for some constant $C$ not depending on $\epsilon$. 

The distance $d$ should satisfy  some **concentration bound** implying, looseley speaking, that the distance between $\mu$ and the empirical measure obtained by $N$ i.i.d. samplings of $\mu$ approaches 0, as $N\rightarrow+\infty$.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see [**this question**][1]; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.


  [1]: https://mathoverflow.net/questions/438622/an-inequality-relating-ell-1-distance-of-input-and-output-of-a-markov-krnel