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• Temporal Airy beam behavior and interactions in a local nonlinear medium are studied. • Simulations show that the Airy beam propagation is controlled by illumination time. • For saturated nonlinearity, Airy beam amplitude controls its propagation direction. • Breathing solitons and parallel soliton pairs are produced in- and out-of-phase. We theoretically examine the temporal behavior of Airy beams and their interactions in photorefractive medium with the split-step Fourier method. Simulations indicate that Airy beam propagations are effectively controlled by the illumination time. Under saturated nonlinearity, the beam intensity maintains the soliton profile at the output section, and its propagation direction is modulated by the beam amplitude. Moreover, the temporal behavior of Airy beam interactions varies greatly with illumination time, especially for the in-phase case. Breathing solitons and parallel soliton pairs are produced in-phase and out-of-phase, respectively. These results are potentially useful in all-optical device applications. [ABSTRACT FROM AUTHOR]