The present article investigates frictional contact mechanics between a moving punch possessing sharp ends and a finite-thickness functionally graded layer resting on the rigid foundation. The rigid punch has either flat or semi-circular profiles, and it moves over the functionally graded layer at a steady subsonic speed. The tangential load developed by the moving punch is proportional to the normal applied load since Coulomb's friction law is adopted. Variations of shear modulus and mass density through the thickness of the layer are expressed by exponential functions, whereas the variation of the Poisson's ratio is neglected. Governing partial differential equations are derived considering two-dimensional theory of elastodynamics, and these equations are solved analytically utilizing Galilean and Fourier transformations. Equations are reduced to a singular integral equation of the second kind. Singular integral equation is solved, and results of analytical approach are compared with those obtained by means of the finite element method in elastostatic case. A very good agreement is achieved between analytical and computational results. Parametric analyses are performed to observe the influences of punch speed, material inhomogeneity and coefficient of friction upon contact stresses of a finite-thickness graded layer resting on the rigid foundation. Stresses observed for relatively thin softening layers are much sensitive to the variation in punch speed.