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We have investigated how electronic excitations that couple via short-range interaction, i.e., triplet excitations and charge carriers, move in a disordered organic semiconductor. In this systematic study, we paid special emphasis to the transition from quasi-equilibrium to nonequilibrium transport as the temperature is lowered from 300 to 10 K. As a method, we used Monte Carlo simulations employing both Marcus as well as Miller&-Abrahams (MA) transition rates. The simulation parameters are the degree of static energetic disorder, the geometric reorganization energy, and the degree of electronic coupling among the hopping sites. In the case of conjugated polymers, the effects of intrachain versus interchain transport are taken into account. In the simulations, we monitor the spectral relaxation of excitations as well as their diffusivity. We find that, below a disorder controlled transition temperature, transport becomes kinetically frustrated and, concomitantly, dispersive. In this temperature regime, transport is controlled by single phonon tunneling, tractable in terms of MA rates, while in the high temperature regime multiphonon hopping, described by Marcus rates, prevails. The results also provide a quantitative assessment of dispersive excitation transport within the intermediate temperature regime in which no analytic theory is available so far. Quantitative agreement between simulation and previous experiments allows one to extract system parameters such as the minimum hopping time and to delineate the parameter range in which Marcus and MA rates should be used in transport studies.