Measurements of the transient photocurrent $I(t)$ in an increasing number of inorganic and organic amorphous materials display anomalous transport properties. The long tail of $I(t)$ indicates a dispersion of carrier transit times. However, the shape invariance of $I(t)$ to electric field and sample thickness (designated as universality for the classes of materials here considered) is incompatible with traditional concepts of statistical spreading, i.e., a Gaussian carrier packet. We have developed a stochastic transport model for $I(t)$ which describes the dynamics of a carrier packet executing a time-dependent random walk in the presence of a field-dependent spatial bias and an absorbing barrier at the sample surface. The time dependence of the random walk is governed by hopping time distribution $\ensuremath{\Psi}(t)$. A packet, generated with a $\ensuremath{\Psi}(t)$ characteristic of hopping in a disordered system [e.g., $\ensuremath{\Psi}(t)\ensuremath{\sim}{t}^{\ensuremath{-}(1+\ensuremath{\alpha})}$, $0l\ensuremath{\alpha}l1$], is shown to propagate with a number of anomalous non-Gaussian properties. The calculated $I(t)$ associated with this packet not only obeys the property of universality but can account quantitatively for a large variety of experiments. The new method of data analysis advanced by the theory allows one to directly extract the transit time even for a featureless current trace. In particular, we shall analyze both an inorganic ($a\ensuremath{-}{\mathrm{As}}_{2}{\mathrm{Se}}_{3}$) and an organic (trinitrofluorenone-polyvinylcarbazole) system. Our function $\ensuremath{\Psi}(t)$ is related to a first-principles calculation. It is to be emphasized that these $\ensuremath{\Psi}(t)$'s characterize a realization of a non-Markoffian transport process. Moreover, the theory shows the limitations of the concept of a mobility in this dispersive type of transport.