In this study, elastodynamic contact mechanics analysis between a moving rigid punch and a functionally graded coating/homogenous substrate system is carried out. Material properties of the substrate are not assumed to be equal to those of the utilized material at the coating interface. A parameter called as interface stiffness ratio controls whether material properties are continuous or not at the interface. Analytical method is developed based on the singular integral equation technique without considering limitation on the continuity on material properties at interface. Partial differential equation system for the functionally graded coating and homogenous substrate are solved analytically by means of Galilean and Fourier transformations. After applying boundary and interface continuity conditions to the mixed boundary value problem, contact problem is reduced to a singular integral equation of the second kind which is solved numerically using a suitable expansion-collocation technique. Computational model is constructed through finite element method and a high level of accuracy is attained between analytically and computationally obtained results. Change in the interface stiffness ratio considerably alters contact stresses in softening coatings especially at higher punch speeds. The importance of interface stiffness ratio is emphasized through examining its influences on elastodynamic contact stresses.