The design and engineering of composite materials is one strategy to satisfy the materials needs of systems with multiple orthogonal property requirements. In the case of rechargeable batteries with lithium metal anodes, the system requires a separator with fast lithium ion transport and good mechanical strength. In this work, we focus on the system polystyrene-block-poly(ethylene oxide) (SEO) with bis(trifluoromethane)sulfonimide lithium salt (LiTFSI). Ion transport occurs in the salt-containing poly(ethylene oxide)-rich domains. Mechanical rigidity arises due to the glassy nature of polystyrene (PS). If we assume that the salt does not interact with the PS-rich domains, we can describe ion transport in the electrolyte by three transport parameters (ionic conductivity, κ, salt diffusion coefficient, D, and cation transference number, t+0) and a thermodynamic factor, Tf. By systematically varying the volume fraction of the conducting phase, ϕc between 0.29 and 1.0, and chain length, N between 80 and 8000, we elucidate the role of morphology on ion transport. We find that κ is the strongest function of morphology, varying by three full orders of magnitude, while D is a weaker function of morphology. To calculate t+0 and Tf, we measure the current fraction, ρ+, and the open circuit potential, U, of concentration cells. We find that ρ+ and U follow universal trends as a function of salt concentration, regardless of chain length, morphology, or ϕc, allowing us to calculate t+0 for any SEO/LiTFSI or PEO/LiTFSI mixture when κ and D are known. The framework developed in this paper enables predicting the performance of any block copolymer electrolyte in a rechargeable battery.