We find a striking trend reversal in the relaxation dynamics of mixtures with strong dynamical asymmetry. Simulations by both Brownian and Newtonian dynamics reveal that in mixtures of fast and slow hard spheres, above a critical density, the dynamics becomes slower upon increasing the mobility of the fast particles. Below that density, the same increase in mobility speeds up the dynamics. The critical density itself can be identified with the glass transition of the mode-coupling theory that does not depend on the dynamical asymmetry. The asymptotic dynamics close to the critical density is universal, but strong pre-asymptotic effects prevail in particular when the dynamical asymmetry also involves size asymmetry. Our observations reconcile earlier findings, where a strong dependence on kinetic parameters was found for the glassy dynamics, with the paradigm that the glass transition is determined by the properties of configuration space alone.