Suppose $A$ is a $m\times n$ Markov matrix, and $C$ is a $m\times k$ Markov matrix. How to decide (analytically or numerically) whether there is a $n\times k$ Markov matrix $B$ such that $AB=C$? I feel that it is a question of feasibility in linear programming.
You could solve this as a Linear Programming feasibility problem.
A Markov Matrix is also known as a stochastic matrix.
So the constraints are:
All elements of B are nonnegative.
All rows sums of B equal 1.
$AB = C$
Edit: Adding CVX program as an easy way to formulate and solve this under MATLAB, given OP comments below.
cvx_begin variable B(n,k) B >= 0 sum(B,2) == 1 A*B == C cvx_end
This will either be determined to be infeasibile, or it will be solved to "optimality", in which case
B will contain a feasible solution after CVX concludes.