Based on the competitive nonlinear fractional Schrdinger equation, the propagation dynamics of single Gaussian beam and double Gaussian beams in nonlocal nonlinear media with cubic-quintic competition are numerically simulated by the split-step Fourier transform method. The results show that the amplitude and width of the beam have a certain effect on the evolution of the Gaussian beam into a bound soliton. For the single Gaussian beam, when the beam amplitudeor width decreases, the pulse power of the bound soliton also decreases. When other conditions are the same, the peak power of the soliton in the local medium is obviously greater than the peak power in the nonlocal medium. When the amplitude of double Gaussian beams decrease or the width increase, the soliton changes from respiratory like state to bound state, and the diffraction phenomenon increase. In local and nonlocal media, both single Gaussian beam and in-phase double Gaussian beams have certain diffraction phenomena, and the diffraction phenomenon is more obvious in nonlocal media. Moreover, the balance between the cubic and quintic nonlocal nonlinearities is also crucial to the stability of the soliton. For the interaction between double Gaussian beams, the interval between the two beams also has a certain influence on the stable transmission of the soliton. In the transmission process, the in-phase double Gaussian beams show a strong attraction. The smaller the beam interval, the greater the interaction force, the weaker the diffraction phenomenon and the smaller the respiratory cycle. And affected by the interaction force, the beam has a certain distance interval in the transmission process. The pulse width and period in nonlocal media are larger than those in local media. While out-of-phase double Gaussian beams exhibit repulsion, the larger the beam spacing, the smaller the repulsive force. The pulse width in local media is wider than that in nonlocal media, and other phenomenan are consistent.