The controllable self-accelerating and self-decelerating Airy–Ince–Gaussian (CAiIG) wave packets in strongly nonlocal nonlinear media are investigated theoretically and numerically by solving the (3 + 1)D Schrödinger equation in cylindric coordinates. Typical examples of the obtained solutions are based on the ratio of the input power and the critical power as well as the initial velocity and the ellipticity. The CAiIG wave packets are constructed by the Airy pulses with initial velocity in temporal domain and the controllable Ince–Gaussian (IG) beams in spatial domain. Self-decelerating and self-accelerating CAiIG wave packets are obtained by selecting different initial velocities. The CAiIG wave packets keep approximately nondispersion properties in temporal dimension. However, the width of packets periodically oscillate or keep steady in spatial dimension according to the ratio of the input power and the critical power. The transverse field patterns of the CAiIG wave packets are manipulated by the IG wave packets, meanwhile the Airy pulses affect their transverse amplitudes. For the self-accelerating CAiIG wave packets, the transverse intensity decays during the propagation. However, to the self-decelerating CAiIG wave packet, the transverse intensity initially vibrates then decays. The direction of the energy flow of CAiIG wave packets in spatial domain periodically changes from the propagating center if the ratio of the input power and the critical power is not equal to 1, which causes the CAiIG beams periodically oscillating during propagation. Their Poynting vector snapshots at different propagating distances are shown.