The propagation and interaction properties of Airy-Gaussian beams in saturable nonlinear medium under fractional effect are investigated numerically by means of split-step Fourier transform method. It is shown that for the propagation of a single Airy-Gaussian beam, variation in the evolution structure of the light field is determined by the fractional Lévy index and the saturable nonlinear strength parameter. The breathing soliton can be formed when the light field evolves under the action of focusing nonlinearity, and the characteristic parameters of the soliton, such as the peak intensity and width, can be controlled by adjusting the Lévy index, the saturable nonlinear strength parameter, the incident light field distribution factor and the light field attenuation coefficient. For the interaction of double Airy-Gaussian beams, two in-phase Airy-Gaussian beams attract each other to generate a single breathing soliton, and two out-of-phase Airy-Gaussian beams repel each other to form a breathing soliton pair. The magnitude of the force between the two beams and the peak intensity of the soliton or soliton pair can be manipulated by adjusting the initial beam interval, Lévy index, the saturable nonlinear strength parameter and the incident light field distribution factor.