In this paper, a new theoretical model of Laguerre-Gaussian power-exponent-phase-vortex beams (PEPVBs) was established, and the theoretical propagation model of the Laguerre-Gaussian PEPVB under paraxial approximation condition was established by using the generalized Collins formula. Then, the relationships among the transmission and focusing characteristics, radial order, topological charge, power index, and transmission distance of Laguerre-Gaussian PEPVBs in free space are simulated by using MATLAB. Research results show that the Laguerre-Gaussian PEPVB and its transmission characteristics are not only related to the power exponent and topological charge, but also related to the radial order of the Laguerre-Gaussian polynomial. And as the transmission distance increases, the beam energy rotates and gathers around the ring. This lays the groundwork for optically manipulating particles to make them move along curved paths and avoid obstacles, which has great significance to the theory and application of new light field control.