This study develops closed-form time-domain solutions for the dynamic response of elastically restrained orthotropic rectangular plates resting on Pasternak foundations and subjected to moving distributed loads and moving distributed masses. The formulation accounts simultaneously for orthotropy, elastic edge restraints, foundation stiffness, and rotatory inertia, thereby providing a unified framework for investigating load–structure interaction in supported plate systems. Starting from the coupled governing fourth-order partial differential equations, a generalized Galerkin procedure is employed to reduce the continuous system to a set of coupled second-order ordinary differential equations. A modified Struble asymptotic technique, combined with Laplace-transform analysis, is subsequently used to derive explicit time-domain response expressions. The resulting solutions provide direct insight into modal contributions, and resonance conditions. Parametric analyses demonstrate that the rotatory-inertia correction, Winkler modulus, and Pasternak shear modulus can substantially modify the amplitude, frequency characteristics, and resonance behavior of the plate. A central finding is that representing the transported body as a moving distributed mass, with its associated inertia and inertial coupling retained, can produce response characteristics markedly different from those predicted by the conventional moving-force idealization. The proposed closed-form solutions therefore provide an efficient analytical tool for parametric assessment of moving-load plate dynamics.