© 2022 Elsevier LtdThis article presents analytical solution procedures for moving Hertzian contact problems involving multi-layer and functionally graded coatings. A frictional circular punch is assumed to slide down the surface of the coating with a specified velocity. Formulation is based on wave equations of plane elasticity, and reduced to a singular integral equation by Galilean and Fourier transformations. Developed procedures are verified through comparisons to the findings available in the literature. Numerical results illustrate influences of factors such as dimensionless punch speed, coefficient of kinetic friction, property variation profile, and contact length-to-thickness ratio upon the contact stresses, and the required contact force.