Lyotropic liquid crystals characterized by a bicontinuous cubic phase (BCP) have a structure characterized by interpenetrated water channels following triply periodic minimal surfaces, which can be stable in excess water conditions and thus suitable in a multitude of applications. The control of the water channels size in these systems has a direct impact on their use for drug delivery, crystallization, and membrane separation processes. In this work we carry out systematic diffusion studies to show how the control on the water channel dimensions directly correlates with the release and separation performance of bicontinuous cubic phases. Specifically, we tune the water channels diameter of the monolinolein/water system by adding different amounts of sucrose stearate, which, having hydration-enhancing properties, can shift the boundaries of the phase diagram. We then design a model bicontinuous cubic phase lipidic membrane of the Im3m space group, having a sugar ester to monolinolein ratio of 20%, and we follow the diffusion within its water channels, by using molecules that differ systematically in size and molecular conformation, and we demonstrate, for each class of molecules, a diffusion-enhanced process upon increase of the water channel diameter. Finally, we also show the ability of the bicontinuous cubic phase to efficiently and selectively separate nanoparticles of a target size, by choosing an amount of sucrose stearate for which the water channel diameter and the nanoparticle dimensions match, demonstrating the possible use of these systems as filtering membranes of tunable molecular cutoff.