Motivated mainly by infrastructure-network management problems, our group has been pursuing analysis and design of various models for network dynamics, which vary in their specifics but broadly can be viewed as either stochastic flow or synchronization processes defined on a graph. So as to obtain a common framework for these models, here we introduce broad and complementary models for linear stochastic flow and synchronization dynamics in networks, that are structured only in that the network's state evolution is Markov and conditionally linear. We first provide mathematical and graphical formulations for each model, and then verify that the models are broad enough to capture several common synchronization/flow networks. As a first analysis, graph-theoretic characterizations of these models' asymptotics are given; these results generalize and enhance known graphical characterizations of existing synchronization/flow models. A comparison of the stochasticity of different flow network models within the framework is also included.