The analytical expression for the propagation of an Airy vortex beam through a paraxial ABCD optical system was derived and used to study propagation dynamics, Poynting vector and vortex trajectory of an Airy vortex beam in gradient-index (GRIN) media, where the effects of vortex core on the main lobe of Airy beam and Poynting vector are stressed. The results show that vortex trajectory and ballistic dynamics are no longer conventional parabolic but trigonometric shapes in the GRIN media. Especially, the Airy vortex beam represents singular behavior at propagation distance z=(2j+1)L/4, and the Poynting vector gradually exhibits reversal before and after z=(j+1)L/2. Based on the analysis of ballistic dynamics, the main lobe of Airy beam is overlapped by vortex core at critical distance, as the beam propagates further, it does not only show self-reconstruction due to the self-healing property of the Airy beam, but also exhibit rotating about the closest vortex.