The master equation for a first-order Markovian process on a state space comprising the positive orthant of a hypercubic lattice arises in the study of several topics of interest in biology. One domain to which this mathematical structure applies is population dynamics where the state variables represent types of organisms and state transition probabilities represent likelihoods of type transformations that may result from mutation, horizontal gene transfer, and the like. Another domain of interest at a lower level in the biological hierarchy is chemical reaction networks comprising metabolic pathways of organisms. While it is common to treat both of these domains with particular mean-field models, it is helpful both to consider the underlying stochastic dynamics and to do so in a manner that unifies model classes. Here we present methods based upon asymptotic matching of perturbative approximation methods we refer to as the interaction term expansion and the system size expansion that are together enable the analysis of arbitrary stochastic reaction networks where the state space is organized as previously stated. We make use of these methods to investigate the dynamics associated with the appearance of new species, whether they be of chemical or whole-organism type, that may be connected to the preexisting network in an arbitrary manner.