This paper presents a stochastic solution procedure for the calculation of the non-stationary freefield response due to a moving load with a random amplitude. In this case, a non-stationary autocorrelation function and a time-dependent spectral density are required to characterize the response at a fixed point in the freefield. The non-stationary solution is derived from the solution in the case of a moving load with a deterministic amplitude. It is shown how the deterministic solution can be calculated in an efficient way by means of integral transformation methods if the problem geometry exhibits a translational invariance in the direction of the moving load. A key ingredient is the transfer function between the source and the receiver that represents the fundamental response in the freefield due to an impulse load at a fixed location. The solution in the case of a moving load with a random amplitude is formulated in terms of the double forward Fourier transform of the non-stationary autocorrelation function. The solution procedure is illustrated with an example where the non-stationary autocorrelation function and the time-dependent standard deviation of the freefield response are computed for a moving harmonic load with a random phase shift. The results are compared with the response in the deterministic case.