Multistate dynamical processes on networks, where nodes can occupy one of a multitude of discrete states, are gaining widespread use because of their ability to recreate realistic, complex behavior that cannot be adequately captured by simpler binary-state models. In epidemiology, multistate models are employed to predict the evolution of real epidemics, while multistate models are used in the social sciences to study diverse opinions and complex phenomena such as segregation. In this paper, we introduce generalized approximation frameworks for the study and analysis of multistate dynamical processes on networks. These frameworks are degree-based, allowing for the analysis of the effect of network connectivity structures on dynamical processes. We illustrate the utility of our approach with the analysis of two specific dynamical processes from the epidemiological and physical sciences. The approximation frameworks that we develop, along with open-source numerical solvers, provide a unifying framework and a valuable suite of tools for the interdisciplinary study of multistate dynamical processes on networks.