This paper presents a general theory of dynamic frictional contact of elastic coatings pressed against by a rigid punch moving with a constant speed. Governing equations of elastodynamics are solved by applying Galilean and Fourier transformations. The contact problem is then reduced to a singular integral equation, which is solved numerically. Developed procedures are verified through comparisons made to the available computational and analytical results. Parametric analyses illustrate the influences of punch speed, material and geometric parameters, and friction on contact stresses. Especially at higher punch speeds, the difference between contact stress magnitudes obtained through elastostatic and elastodynamic solutions is rather significant. A formulation based on the elastodynamic theory is required to compute contact stresses generated in such problems.