Some experiments show that the practical atmosphere deviates from ideal Kolmogorov model. In this paper, based on the extended Huygens-Fresnel principle and the non-Kolmogorov turbulence model proposed by Toselli et al., the analytical expression for the propagation of partially coherent hyperbolic-sine-Gaussian vortex beams through non-Kolmogorov atmospheric turbulence is derived and used to study the dynamic evolutions of composite coherence vortices formed by coherent and incoherent superpositions of two partially coherent hyperbolic-sine-Gaussian vortex beams in non-Kolmogorov atmospheric turbulence. It is shown that the evolution process of the average intensity of the superimposed beam depends on the general exponent of the non-Kolmogorov turbulence, the sign of the topological charge of the superimposed vortex beam in the source plane, and superposition scheme. The motion, the creation and the annihilation of composite coherence vortices may take place upon propagation through non-Kolmogorov turbulence, and the general exponent , sign of the topological charge and superposition scheme affect the evolution behavior. Finally, the results are compared with those of the previous work.