Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution.

The mutual information between the input and the output of $C_1$ must be greater or equal than the mutual information between the input and the output of the composed channel $C_1\circ C_2$ (i.e. act with $C_1$ first then feed the output to $C_2$). This follows from the data processing inequality.

My questions are:

Given $p(x)$, what are the channels $C_2$ for which equality holds (i.e. mutual information is non-decreasing). I know from of proof of data processing inequality that is true if and only if one has a Markov chain, but what can we say about the conditional transition matrices, i.e. relations between $p(x)$, $C_1$ and $C_2$? Or, in other words, what is the functional form of $C_2$ as a function of $C_1$ and $p(x)$?

And the reverse: given $C_2$, what are the input distributions $p(x)$ for which the mutual information is non-decreasing?

I wasn't able to find an elegant solution to this problem, I have only some partial solutions. For example, if $C_2$ is a permutation channel, then mutual information stays the same no matter what $p(x)$ is. Thanks!

PS: I hope it is clear what I mean by mutual information between the input and output of a channel, it is the mutual information of the joint probability distribution obtained by multiplying the elements of the transition matrix with the corresponding component of the input, $p(x,y)=p(y|x)p_{0}(x)$.