I think this is basically the inverse question of Matrices whose exponential is stochastic.
i.e. what are sufficient conditions on the matrix representation of an evolution operator of a (finite) discrete Markov chain for it to be embeddable in a continuous Markov chain?
I've found some old paper that may answer this (something about embeddability criteria) but I can't access it as it published in a closed-access journal.
I hope this is a sane question.