The Analytical Transport Network (ATN) model for flow through microstructural networks is extended to linearly coupled flows subject to Onsager reciprocity. Electrokinetic flow is used as an example system. Through the extension, we gain an improved understanding of if, and how, morphology and topology influence coupled flow systems differently than un-coupled flows. In Part 1, a channel-scale model is developed to describe electrokinetic flow through a channel of arbitrary morphology. The analytical model agrees well with finite element analysis (FEA), but is significantly less expensive in terms of computational resources, and, furthermore, offers general insight into morphology's additional influence on coupled flows relative to uncoupled flows. In Part 2, we exploit these savings to develop a computationally economical, network-scale model and associated algorithm for its implementation to voxel-based 3D images. Included in the algorithm is a means for rapidly calculating a structure's tortuosity factor. This modeling effort represents an important initial step in extending the ATN approach to coupled flow phenomena relevant to emerging technologies that rely on heterogeneous, hierarchical materials.