We numerically investigate the propagation properties of ring Airy Gaussian beams (RAGBs) with cosine modulated optical vortices (CMOV). In comparison to the common RAGBs without any modulation, the dynamic propagation of RAGBs with CMOV exhibits a unique feature: the rings of RAGBs with CMOV will gradually shrink into several main lobes with the increase of the propagation distance. The number of lobes and the peak intensity of each lobe are determined by the factors of cosine modulated function. By designing the initial phase, we can easily change the transversal location of the peak intensity. Our results may find potential applications in optical manipulations.