Recent experiments have reported that the lamellar phase of salt-doped tapered copolymers exhibit higher ionic conductivity compared to those seen in similar morphologies of diblock copolymers. Such observations were in turn rationalized by invoking the corresponding glass transition temperature of the segregated copolymers. In this work we report the results of coarse-grained molecular dynamics simulations to identify the mechanisms underlying such characteristics. Explicitly, we probe the combined influences of the degree of segregation and the disparity in mobilities of the segments of the two blocks, upon the local relaxation dynamics of tapered copolymers segregated in lamellar phases. Our results show that the local dynamics of tapered copolymers depend on two independent factors, viz., the degree of segregation of such copolymers relative to their order-disorder transition temperature, and the relative mobilities (glass transition temperatures) of the two blocks. In qualitative correspondence with experiments, we find that for appropriate combinations of mobility ratios and degree of segregation, the lamellar phases of tapered copolymers can exhibit faster local segmental dynamics compared to diblock copolymers.