This study investigates the dynamic behaviour of an elastically supported uniform Rayleigh beam subjected to the passage of moving distributed masses with varying velocities, where the loading conditions vary arbitrarily with time. The motion of this problem is described by a fourth-order partial differential equation, which governs its behaviour. The beam’s non-stationary response under such dynamic loading scenarios is analysed using the weighted residual method, which converts the governing equation into a sequence of linked second-order differential equations to facilitate the analysis. A rewritten version of Struble’s asymptotic method further simplifies the transformed governing equation. This modification aids reduction in the complexity of the equation. The closed-form response is contrasted for the acceleration and deceleration motion. The study thoroughly examines how different velocities and frequencies of the moving force affect the dynamic behaviour of the beam. Key aspects explored include the influence of axial force, foundation modulus, and shear modulus in the support structure, the impact of varying mass distributions, and the time-dependent nature of the applied loads. The results help further understand the structural dynamics in complex environments and offer insights into optimising the design and performance of similar systems under non-stationary dynamic loads.