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The problem of eliminating fast-relaxing variables to obtain an effectivedrift-diffusion process in position is solved in a uniform and straightforwardway for models with velocity a function jointly of position and fast variables.A more unified view is thereby obtained of the effect of environmentalinhomgeneity on the motion of a diffusing particle, in particular, whether adrift is induced, covering both passive and active particles. Infinitesimalgenerators (equivalently, drift-diffusion fields) for the contracted processesare worked out in detail for several models.

Drift-diffusion processes from elimination of fast variables under inhomogeneous conditions