Based on the complex cubic-quintic Ginzburg-Landau equation with high-order effects, we numerically investigated the propagation characteristics of bound state solitons and pulsating-like solitons which are generated by Airy pulses by using the split-step Fourier method. The results show that the transmission directions of the two solitons are deflected to the right under the influence of self-frequency shift effect and self-steepening effect, and the deflect speed can be controlled by adjusting the parameters. In addition, the peak power of the bound state solitons decreases with the increase of the self-frequency shift effect. However, the change of self-steepening effect has little effect on the peak power of the bound state solitons. The third-order dispersion effect will also cause the bound state solitons to deflect to the right or left, but the direction of the deflection is determined by the sign of the third-order dispersion effect. For the pulsating-like solitons, the addition of the self-frequency shift effect will deflect them to the right, and the deflect speed increases with the increase of the self-frequency shift effect, and the peak power decreases with the increase of the self-frequency shift effect. However, when the higher-order effect exceeds a certain critical value, the propagation characteristics of the pulsating-like solitons will change abnormally. The pulsating-like solitons will be transformed into fixed-shape solitons under the influence of the self-frequency shift effect; the self-steepening effect will affect the stability of the solitons and deflect the propagation of pulsating-like solitons to the left, while the third-order dispersion effect makes pulsating-like solitons evolve into rectangular-like pulses, and the peak power of pulsating-like solitons decreases with the increase of higher-order effects. The relevant results not only enrich the propagation dynamics of Airy beams, but also have certain potential applications in the manipulation of Airy pulses. At the same time, it will also provide a certain theoretical reference for research in the fields of light guides and optical switches.