We perform a time-dependent study of the driven dynamics of overdamped particles which are placed in a one-dimensional, piecewise linear random potential. This set-up of spatially quenched disorder then exerts a dichotomous varying random force on the particles. We derive the path integral representation of the resulting probability density function for the position of the particles and transform this quantity of interest into the form of a Fourier integral. In doing so, the evolution of the probability density can be investigated analytically for finite times. It is demonstrated that the probability density contains both a $delta$-singular contribution and a regular part. While the former part plays a dominant role at short times, the latter rules the behavior at large evolution times. The slow approach of the probability density to a limiting Gaussian form as time tends to infinity is elucidated in detail.
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