The uniform size and complex chemical topology of the pores formed by self-assembled amphiphilic molecules such as liquid crystals make them promising candidates for selective separations. In this work, we observe the transport of water, sodium ions, and 20 small polar solutes within the pores of a lyotropic liquid crystal (LLC) membrane using atomistic molecular simulations. We find that the transport of a species is dependent not only on molecular size but also on chemical functionality. The membrane's inhomogeneous composition gives rise to radially dependent transport mechanisms with respect to the pore centers. We observe that all solutes perform intermittent hops between lengthy periods of entrapment. Three different trapping mechanisms are responsible for this behavior. First, solutes that drift out of the pore can become entangled among the dense monomer tails. Second, solutes can donate hydrogen bonds to the monomer head groups. Third, solutes can coordinate with sodium counterions. The degree to which a solute is affected by each mechanism is dependent on the chemical functionality of the solute. Using the insights developed in this study, we can begin to think about how to redesign existing LLC membranes to perform solute-specific separations.