The study is focussed on the near-critical regimes of the moving load on a coated elastic half-space. The material properties of the coating are assumed to be depending on the vertical coordinate. The analysis relies on the hyperbolic-elliptic formulation for the Rayleigh wave induced by prescribed surface loading, containing an elliptic equation for the elastic potential governing its decay over the interior, and a singularly perturbed wave equation on the interface between the layer and the substrate. This scalar formulation allows significant simplifications, including in particular, classification of the near-resonant regimes. The method of fictitious absorption is then used in order to incorporate the effect of poles, associated with the radiation of energy from the moving source. Finally, numerical illustrations of results for several types of vertical inhomogeneity are presented.