Block copolymers self-assemble into a variety of morphologies that can serve as templates for preparing conductive or porous materials such as batteries and membranes. These morphologies can be either thermodynamic phases at equilibrium or defective nonequilibrium states that are kinetically trapped. It is an important engineering challenge to predict key material properties from an underlying morphology, particularly when the material may have inherent defects from processing. Here, we focus on the self-diffusion of a tracer through porous membranes assembled from model triblock copolymers. We generate a large library of both equilibrium and nonequilibrium membrane structures using self-consistent field theory, and we simulate solute diffusion in these structures as a simple random walk on a lattice. We show that the solute self-diffusion coefficient strongly correlates with two of the Minkowski functionals characterizing the morphology of the pores: their volume and their integrated mean curvature. We find that, relative to the corresponding equilibrium morphologies, the structure and transport properties associated with nonequilibrium morphologies of gyroid-forming polymers are more tolerant of defects than those of lamella-forming polymers. However, the nonequilibrium morphologies of lamella-forming polymers exhibit a rich diversity of pore structures and corresponding diffusivities, which may prove helpful for designing membranes with targeted properties.