Partially coherent beams with nonconventional correlation functions have been extensively studied due to their wide and important applications in free-space optical communication, particle trapping, image transmission and optical encryption. Here, we study the propagation of nonuniform cosine-Gaussian correlated Bessel-Gaussian beam (cGBCB) in detail. Analytical expressions for the cross-spectral density function of cGBCBs through paraxial ABCD system are derived based on the extended Huygens-Fresnel integral. By use of the derived formulae, the intensity distribution properties of a nonuniform cGBCB on propagation in free space are analytically investigated. Some numerical calculation results are presented and discussed graphically. It is found that when the root-mean-square correlation width δ and the parameter controlling the degree of coherence profiles β are appropriately chosen, the intensity distribution of the nonuniform cGBCB displays self-splitting properties during propagation. We point out that rather than a simple duplication, the self-splitting behaviour consists of a complex process in which the dark hollow pattern for cGBCB is gradually filled in the centre at first, then starts to split with increasing the propagation distance, and most impressively, an evolution process from a single dark hollow beam in the source plane to quadruple dark hollow profiles in certain propagation ranges can be realized. The influence of correlation function on the evolution properties of the intensity distribution is investigated, demonstrating that the values of parameters δ and β of the correlation function play a critical role in inducing the self-splitting effect for nonuniform cGBCB on propagation in free space. Therefore, it is clearly shown that modulating the correlation function of a partially coherent beam can alter the coherence length and the degree of nonuniformity, and thus provides an effective way of manipulating its propagation properties. We also find the evolution speed of the intensity distribution can be greatly affected by the topological charge n of the beam function and the parameter R controlling the hollow size of cGBCB in source plane, e. g. the intensity distribution evolves into quadruple dark hollow profiles more slowly with larger n or smaller R. As is well known, the dark-hollow intensity configurations are useful in many applications and have been extensively studied both theoretically and experimentally. Therefore, the results drawn in the paper develop an alternative way to realize dark-hollow beam array, and further pave the way for dark hollow beam applications in long-distance free-space optical communications.