Problems on motion with a constant subseismic velocity of an oscillating load on the boundary of an isotropic elastic layer inhomogeneous over the thickness are studied in a three-dimensional formulation. Quantitative estimates are given for the upper limits on the magnitudes of the velocity of motion and the load vibration frequency for which a unique solution exists for the problem in energy classes. In cases when no energy solution exists, principles are formulated to extract the unique solution and a solution is given in the far field. Results are presented of numerical computations of the waves field characteristics in the cases of the motion of a normal concentrated load in a homogeneous layer. Situations are noted in which a different number of waves propagates in different layer domains. The problems considered are of interest for seismology and in designing aerodrome coverings.