Systems where multiple trapping or other manifestations of disorder lead to a non-local temporal evolution which results in the macroscopic observation of anomalous kinetics, are shown to exhibit a non-exponential, Mittag-Leffler decay in Kramers-type escape problems. The detailed behaviour of the survival probability is studied, and it is demonstrated that the associated escape rate is time dependent, exhibiting a turnover from a self-similar to a logarithmic pattern. Comparisons to experiment, and to local models are drawn.