What is often referred to as a Hertzian contact can undergo plasticity either at the macroscale, due to an accidental overload, or at an asperity scale, due to the presence of surface defects and/or roughness. An elastic solution does not explicitly consider the surface velocity or loading history, but it is also apparent that a moving (rolling) load will not yield the same residual stress and strain distribution as a purely vertical loading/unloading. Three-dimensional (3D) analysis is also more complex than the two-dimensional (2D) problem because it implies a change in the surface conformity. This paper presents the results of a numerical investigation of frictionless elastic-plastic elliptical point contacts with a moving load, as compared to a purely vertical (indentation) load. In the present analysis, both bodies may behave in an elastic-plastic mode. Both kinematic and isotropic hardening are considered to account for repeated rolling contacts. The contact pressure and the plastic strain are found to be reduced when the two bodies are elastic-plastic, as compared to the case in which one of the bodies remains elastic. Numerical results also indicate that at a given load intensity, the maximum contact pressure and equivalent plastic strain are affected by the contact geometry (circular and elliptical point contacts) and differ significantly when the load is moving as compared to purely vertical indentation. Although the maximum elastic contact pressure (Hertz solution) is often used as a control parameter for rolling contact fatigue analysis, whatever the geometry of the contact (point, elliptical, or line contact), the results presented here show that the effective contact pressure and subsequent residual strains are strongly dependent on the contact geometry in the elastic-plastic regime.