We investigate a family of the cosh-Pearcey-Gaussian vortex (CPeGV) beams, obtain the general propagation expressions of a CPeGV beam, and study the longitudinal and transverse Poynting vector and angular momentum density (AMD) when the CPeGV beams propagate in uniaxial crystals. The effects of the cosh modulation parameter, topological charge, and propagation distance on the propagation properties of CPeGV beams are discussed. A larger cosh modulation parameter can lead the energy transfer significantly along the transverse Poynting vector direction. Moreover, we also investigate how the cosh modulation parameter and topological charge influence the propagation properties in the far-field. A larger cosh modulation parameter can lead AMD to present four-lobes structures rather than usual parabolic curves. Our investigation will provide a better understanding of the state of the CPeGV beams propagating in uniaxial crystals and be useful for applications in information transmission.