The propagation of decelerating Airy pulses in non-instantaneous cubic medium is investigated both theoretically and numerically. In a Debye model, at variance with the case of accelerating Airy and Gaussian pulses, a decelerating Airy pulse evolves into a single soliton for weak and general non-instantaneous response. Airy pulses can hence be used to control soliton generation by temporal shaping. The effect is critically dependent on the response time, and could be used as a way to measure the Debye type response function. For highly non-instantaneous response, we theoretically find a decelerating Airy pulse is still transformed into Airy wave packet with deceleration. The theoretical predictions are confirmed by numerical simulations.