The dynamics of the Pearcey-Gaussian beam with Gaussian potential in the fractional Schrödinger equation are investigated. In the free space, varying the Lévy index offers a convenient way to control the splitting and bending angle of the beam. In the presence of Gaussian potential, with the increasing of propagation distance, the process is repeated in a breath-like motion. The periodicity also can be changed by adjusting the potential parameter and incident beam arguments, such as potential height, potential width and transverse wavenumber. The transmission and reflection of the beam can also be controlled by varying the potential parameters. Moreover, when a symmetrical Gaussian potential barrier is selected, total reflection is more likely to occur. These unique characteristics show the possibility in controlling the dynamics of Pearcey-Gaussian beam with the fractional Schrödinger equation system.