For a variety of molecularly doped, pendant-, and main-chain polymers, and vapor-deposited molecular glasses, the mobility of photoinjected charges at high electric fields is described by the Poole-Frenkel law; μ = μ₀ exp (γ√E) . Apart from their organic constituents, the primary transport-related feature shared by these materials is the lack of a periodic structure. We review the relation between the √E-dependent mobilities and dispersive transport, as described by the theory of Scher and Montroll for hopping transport in a disordered medium. We show that with a small modification, the theory predicts dispersive transport below and nondispersive transport above a transition temperature T_c. We argue that the √E dependence of the mobility and the universality of the current-time curves may be retained above and below T_c if the bulk film behaves as a lattice of bonds of length L, where L is intermediate between the dopant spacing and the thickness of the sample.