H-1 and F-19 magnetic resonance microscopies are used to determine the characteristics of diffusion in four different network-solvent systems. Time-resolved analysis of the concentration profiles of toluene in polybutadiene and hexafluorobenzene in poly(methyl silicone) demonstrates that solvent transport in these systems is Fickian. The kinetics of solvent transport in two glassy networks, however, is shown to be non-Fickian. These latter two systems are characterized by sharp solvent fronts which propagate into the cores of the samples at a constant velocity. The swelling kinetics are quantified by applying a simple model which couples the kinetics of solvent diffusion to a second-order phase transition which induces network relaxation. Parameterization is accomplished with two kinetic terms and one thermodynamic parameter. These are a mass-fixed glassy diffusion coefficient, a network relaxation constant, and a critical concentration corresponding to the concentration of solvent necessary to induce a glass to rubber transition. Solvent from velocities, obtained through magnetic resonance microscopy, are used with independently derived critical concentrations to calculate the glassy diffusion coefficient and network relaxation rate constant. Kinetic swelling data are then fit with theoretical uptake curves computed using these parameters. A high-quality fit demonstrates that the proposed model successfully quantifies non-Fickian transport using a small number of physically based dynamic parameters.