The evolution of finite array Airy beams (FAAB) propagating in gradient refractive index medium (GRIM) is investigated based on light transfer matrix and generalized Huygens-Fresnel integral equation. Firstly, the general propagation formulae of the Airy beam and FAAB passing through the paraxial optical system are calculated in space domain, respectively. Then the incident FAAB intensity profiles changed with exponential truncation and beam wrist size factors are discussed. The intensity characteristics and the periodicity of the transmission of the FAAB on several cross-sections of the GRIM are researched. The propagation behavioral differences between two kinds of beams are compared. The results are of theoretical significance to the transmission and communication of the array beams, because both the atmospheric random dither and graded index optical fibers can be abstracted as gradient index models.