Supramolecular assembly allows for enhanced control of bulk material properties through the fine modulation of intermolecular interactions. We present a comprehensive study of a cross-linkable amphiphilic wedge molecule based on a sulfonated trialkoxybenzene with a sodium counterion that forms liquid crystalline (LC) phases with ionic nanochannel structures. This compound exhibits drastic structural changes as a function of relative humidity (RH). Our combined structural, dynamical, and transport studies reveal deep and novel information on the coupling of water and wedge molecule transport to structural motifs, including the significant influence of domain boundaries within the material. Over a range of RH values, we employ Na-23 solid-state NMR on the counterions to complement detailed structural studies by grazing-incidence small-angle X-ray scattering. RH-dependent pulsed-field-gradient (PFG) NMR diffusion studies on both water and the wedge amphiphiles show multiple components, corresponding to species diffusing within LC domains as well as in the domain boundaries that compose 10% of the material. The rich transport and dynamical behaviors described here represent an important window into the world of supramolecular soft materials, carrying implications for optimization of these materials in many venues. Cubic phases present at high RH show fast transport of water (2 X 10(-10) m(2)/s), competitive with that observed in benchmark polymeric ion conductors. Understanding the self-assembly of these supramolecular building blocks shows promise for generating cross-linked membranes with fast ion conduction for applications such as next-generation batteries.