Diblock copolymers (DBPs) are used in numerous current and potential applications, from composite materials to electrolyte membranes. Depending on the chemical incompatibility of the blocks and their volume fractions, they self-segregate into microdomains inducing anisotropy and different symmetries to the microstructure, resulting in significant variability in the macroscopic properties. DBPs are emerging as a candidate for replacing conventional liquid electrolytes in electrochemical devices, due to their improved chemical and mechanical stability. In contrast to isotropic melts, the self-segregation of DBPs allows the decoupling of ionic transport and mechanical properties. In this work, molecular dynamic simulations of coarse-grained DBPs were devised and carried out to unveil correlations between the microstructure and both ionic mobility ($\mu$), and dynamics/viscoelasticity (directly correlate with mechanical properties) of DBPs. It was found that across different morphologies and chain lengths ($N$), $\mu$ is mainly controlled by the extent of microdomains mixing and the tortuosity of the conductive path. Furthermore, the local fluctuations in the density of the polymer matrix have a non-negligible effect on the transport of ions. Our study of dynamic and viscoelastic properties of DBPs revealed that the interface between the two blocks constrains chain conformations in a way akin to topological constraints caused by entanglements. Specifically, the DBP interface gives rise to a temperature-dependent early crossover from Rouse to reptation scaling of the self-diffusion coefficient with $N$ compared to isotropic melts. Rheologically, the interface manifests into a modulus plateau similar to that of the rubbery plateau of entangled polymer melts that arises at the same Rouse to reptation scaling crossover $N$. Overall, the results of this study shed some light on the effects of various design and operating parameters, such as block composition, temperature, and external forces, on the transport properties of DBPs. It is expected that these results will provide a fundamental basis to complement ongoing experimental and modeling efforts devoted to engineer DBPs for target applications.