The problem of quickest detection of a change in the distribution of a sequence of independent observations is considered. The pre-change distribution is assumed to be known and stationary, while the post-change distributions are assumed to evolve in a pre-determined non-stationary manner with some possible parametric uncertainty. In particular, it is assumed that the cumulative KL divergence between the post-change and the pre-change distributions grows super-linearly with time after the change-point. For the case where the post-change distributions are known, a universal asymptotic lower bound on the delay is derived, as the false alarm rate goes to zero. Furthermore, a window-limited CuSum test is developed, and shown to be asymptotically optimal. For the case where the post-change distributions have parametric uncertainty, a window-limited generalized likelihood-ratio test is developed and is shown to be asymptotically optimal. The analysis is validated through numerical results on synthetic data. The use of the window-limited generalized likelihood-ratio test in monitoring pandemics is also demonstrated.