Laguerre-Gaussian correlated Schell-model (LGCSM) vortex beam is introduced as an extension of LGCSM beam which was proposed [Opt. Lett.38, 91 (2013)Opt. Lett.38, 1814 (2013)] just recently. Explicit formula for a LGCSM vortex beam propagating through a stigmatic ABCD optical system is derived, and the propagation properties of such beam in free space and the focusing properties of such beam are studied numerically. Furthermore, we carry out experimental generation of a LGCSM vortex beam, and studied its focusing properties. It is found that the propagation and focusing properties of a LGCSM vortex beam are different from that of a LGCSM beam, and we can shape the beam profile of a LGCSM vortex at the focal plane (or in the far field) by varying its initial spatial coherence. Our experimental results are consistent with the theoretical predictions, and our results will be useful for particle trapping.