Given two probability measures, which Markov kernels act on the first to give the second? Suppose I have a Markov kernel $\Sigma \colon \Omega\to\Delta(S)$.
For any measure $\tau \in \Delta(\Omega)$, I can obtain a measure $\tau\Sigma \in \Delta(S)$ considering
$$(\tau\Sigma)(A) = \int_{\Omega} \Sigma(\omega)(A) \,\mathrm{d}\tau(\omega) \,.$$
In other words, I can obtain a measure on $S$ by considering the average of the kernel with respect to the measure on $\Omega$.
Suppose now I have two probability measures $\mu \in \Delta(\Omega)$ and $\nu \in \Delta(S)$, it makes sense to consider the (maybe empty) set
$$\{\Sigma \colon \Omega \to \Delta(S) \mid \mu\Sigma = \nu\} \,.$$
How can I characterize such a set? When $\Omega$ and $S$ are finite sets, one can represent Markov kernels as matrices (summing to one on coloumns) and probability distribution as vectors, and the problem  reduces to a linear algebra one, but still, I can’t see an easy general solution.
Do you have any reference on the  operation above producing a probability measure from a probability measure and a Markov kernel? I can only notice thatwhen the Markov kernel is something of the form $\Omega \to \Delta(\Omega)$ (a coalgebra for $\Delta$) it seems to define the action of the (monoid?) of such Markov kernels over $\Delta(\Omega)$.
 A: Using the Disintegration theorem, it suffices that there exist a map $\pi : S \rightarrow \Omega $ such that $\mu$ is the pushforward of $\nu$ with respect to $\pi$. Then, for each such $\pi$, the disintegration theorem furnishes you with a unique Markov kernel (up to measure zero).
Now, given measures $\mu$ and $\nu$, does there always exist such a map $\pi$? For general spaces this is a complicated question. However, in the discrete case you are considering, one can always produce such a map provided that the cardinality of the support of $\mu$ divides that of $\nu$ (for instance, the solution to the so-called Monge problem from optimal transport provides such a map).
However, you ask for how to characterize the set of all Markov kernels K for which $ K \mu = \nu $. In principle this set is at least as complicated as the set of all functions $\pi$ that push forward $\nu$ to $\mu$. I expect this set is very complicated unless one puts very strict assumptions on $\mu$ and $\nu$, but am not aware of any general work in this direction.
