The polarization evolution of vector beams (VBs) generated by q-plates is investigated theoretically and experimentally. An analytical model is developed for the VB created by a general quarter-wave q-plate based on vector diffraction theory. It is found that the polarization distribution of VBs varies with position and the value q. In particular, for the incidence of circular polarization, the exit vector vortex beam has polarization states that cover the whole surface of the Poincaré sphere, thereby constituting a full Poincaré beam. For the incidence of linear polarization, the VB is not cylindrical but specularly symmetric, and exhibits an azimuthal spin splitting. These results are in sharp contrast with those derived by the commonly used model, i.e., regarding the incident light as a plane wave. By implementing q-plates with dielectric metasurfaces, further experiments validate the theoretical results.