On the basis of the variable-coefficient fractional Schr?dinger equation with potential barriers, we study the influence of chirp parameters and potential barrier functions on propagation dynamics of Gaussian beams by the combination of the numerical simulation and analytical method. The results reveal that the linear chirp will weaken the intensity of a sub-beam after splitting, while the quadratic chirp will change the respiratory amplitude and width of beams, and thus the beams no longer oscillate strictly according to the sinusoidal law, which leads to a large and a small respiratory trajectories eventually. After the introduction of the barrier effect, the beams at the barriers deform due to reflection and transmission. The reflected and transmitted beams are still modulated by the diffraction coefficient and are superimposed at the barriers before transmitting again. Under an appropriate barrier depth, the beams are completely reflected when they encounter the barriers, and all the energy of the beams is trapped between the two barriers. Thus, in the fractional system, the regulation and management of beam transmission can be achieved through potential barriers and chirp parameters.