We propose the analytic solution of cross-spectral density matrix (CSDM) of a partially coherent vortex beam while propagating. The solution to the propagation law is appropriate for a Gaussian Schell-model vortex beam to any order. It is found that the solution of CSDM of a Gaussian Schell-model beam revealed by Emil Wolf is a special instance. By this general solution, the intensity distribution, especially the degree of polarization and the mean-squared beam width of the partially coherent vortex beam, are investigated. It is found that the propagation character of the beam width is decided by the correlation-length, and that the topological charge affects only the value of the beam width.