Based on the fractional derivative Schrdinger equation with variable coefficients, the propagation and interaction of cosine-Gaussian beams are studied by means of the split-step Fourier algorithm. In the linear case, a single non-chirped cosine-Gaussian beam splits into two identical Gaussian beams during transmission and forms a Y-shaped beam structure, while a chirped cosine-Gaussian beam generates two asymmetric beams during transmission. The value of chirped parameter affects the bifurcation of the cosine-Gaussian beams. The two branches of the cosine-Gaussian beams are asymmetrically distributed as the chirp parameter increases. One branch of the cosine-Gaussian beams disappears completely when the value of the chirp parameter is up to 1.5. When the Lévy index is 1, the shape remains unchanged after the interaction of the two chirped cosine-Gaussian beams. In addition, the cosine-Gaussian beam under the action of fractional diffraction modulated by periodic function, linear function, and power function, the transmission results show periodic structure, parabolic trajectory transmission, and shape-invariant transmission, respectively. In the nonlinear case, the saturating nonlinear effect destroys the periodic structure of the cosine-Gaussian beam while suppressing the propagation broadening of the cosine-Gaussian beam. Especially, the cosine-Gaussian beam can be trapped to form localized modes similar to an optical soliton, which retains its profile during propagations under the action of the saturating nonlinear effect. These results of the research may be used to control the cosine-Gaussian beam trajectory.