I think this is basically the inverse question of Matrices whose exponential is stochasticMatrices 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.