Lyotropic Liquid Crystals (LLCs) are a class of lipid-based membranes with a strong potential for drug-delivery employment. The characterization and control of their transport properties is a central issue in this regard, and has recently prompted a notable volume of research on the topic. A promising experimental approach is provided by the so-called diffusion setup, where the drug molecules diffuse from a feeding chamber filled with water to a receiving one passing through a LLC. In the present work we provide a theoretical framework for the proper description of this setup, and validate it by means of targeted experiments. Due to the inhomogeneity of the system, a rich palette of different diffusion dynamics emerges from the interplay of the different time- and lengthscales thereby present. Our work paves the way to the employment of diffusion experiments to quantitatively characterize the transport properties of LLCs, and provides the basic tools for device diffusion setups with controlled kinetic properties.