Internal charge drift is induced by applying an external electric field on the photorefractive crystal. The drifted charges may lead to the diffusion of spatial charges which accounts for the nonlocal nonlinearity. The propagation of spatial soliton supported in the photorefractive medium with drift and diffusion nonlinearity is investigated theoretically. The effective “acceleration” of soliton is derived by analyzing the dynamics of the (1+1) dimensional spatial soliton in this kind medium with equivalent-particle approximation methods. The parameters of input soliton and nonlocality determine the effective acceleration of soliton. The results can be used to calculate the propagation trajectory of solitons in a certain parameter range. We also simulate the propagation of soliton under perturbation of nonlocal nonlinearity. The theoretical analysis results are in good agreements with the numerical simulation results.