Based on the nonlinear Schrdinger equation with nonlocal nonlinear effects, we studied the propagation dynamics of the finite-energy Airy beams with initial chirp in competing nonlocal nonlinear media by using the split-step Fourier method. The results show that the initial chirp can change the propagation velocity of the single finite-energy Airy beam, and the larger the chirp, the greater the pulse offset velocity. However, the chirp has little effect on the breathing period and amplitude of the beam. When the initial input is double Airy beams, the chirp increases the repulsive force between the two beams in the weak nonlocal medium, thus affecting the breathing period of the solions. In the strong nonlocal medium, the interaction between the two beams is more complex, showing a quasi-periodic characteristic with complex structure, but the chirp leads to the periodic change of the quasi-periodic structure; with the increase of the absolute value of the chirp, the period of the quasi-periodic structure becomes larger, and the beam diffraction is enhanced.