When propagating in free space, the transversal distribution of the degree of polarization of an anisotropic electromagnetic Gaussian-Schell model (AEGSM) beam will generally undergo a complex evolution process. We find that this transversal distribution of the degree of polarization of an AEGSM beam can be controlled by exploiting the partial correlation properties of the source. The main research of our paper falls into two parts. First, the concept of analogical propagation of the transversal distribution of the degree of polarization is proposed, and the condition for an AEGSM beam having an analogical propagation is obtained. When an AEGSM beam is on analogical propagation, the distribution of the degree of polarization on any cross section of the beam is always similar to that on the source plane, except that the size of the distribution pattern will expand continuously as the propagation distance increases. Second, the far-field transversal distribution of the degree of polarization is considered, and the condition for the far-field transversal polarization distribution of an AEGSM beam to be always of circularly symmetric shape, no matter how complicated it is on the source, is obtained. Our research is expected to find applications in areas that make use of the polarization properties of random electromagnetic beams.