The convex partially coherent beam (CPCB) is a special type of nonuniformly correlated beam with a convex-shaped complex degree of coherence (DoC) distributions. Previously our research has illustrated the potential of CPCBs with super-Gaussian DoCs in free-space optical communications (FSOC), mainly manifested as self-focusing which can be transferred into extra scintillation reduction and SNR gain. In this study, the effects of the DoC transition slopes are analyzed and more details about the turbulence propagation of CPCBs with super-Gaussian shaped DoC are revealed. By means of wave optics simulation, the longitudinal intensity evolution of the CPCB is explored, showing that the DoC slope has a profound influence on the self-focusing features such as the focusing plane and the peak intensity. Aperture scintillation and mean SNR at the receiver end of some short-range vertical turbulent links are numerically computed. The obtained results show that, with CPCBs, an ~2 dB SNR gain can be achieved as compared to conventional Gaussian Schell-modal (GSM) beams. However, CPCBs are preferred only in shorter links, which is found to be relevant to the power-in-the-bucket of the receiving aperture. Furthermore, the impacts of the ratio of the source coherence time to the detector integration time are investigated, implying that the CPCB is less susceptible than the GSM. We have also examined the off-axis scintillation of the CPCB. Due to its convex-shaped DoC, the CPCB has significantly reduced off-axis scintillation, which can be beneficial in the presence of pointing errors.