The purpose of this article is to describe various conditions on the parameters of pairs of nonhomogeneous Poisson or pure birth processes under which the corresponding epoch times or interepoch intervals are stochastically ordered in various senses. We derive results involving the usual stochastic order, the multivariate hazard rate order, the multivariate likelihood ratio order, as well as the dispersive and the mean residual life orders. A sample of applications involving generalized Yule processes, load-sharing models, and minimal repairs in reliability theory illustrate the usefulness of the new results.