The equivalence between absorbing and reflecting barrier problems for random walks is shown to hold for stochastically monotone Markov processes. For Markov chains in continuous time this relation is expressed directly in terms of the $Q$-matrices of the chains. Some examples are given.
"The Equivalence of Absorbing and Reflecting Barrier Problems for Stochastically Monotone Markov Processes." Ann. Probab. 4 (6) 914 - 924, December, 1976. https://doi.org/10.1214/aop/1176995936