According to the (3+1)-dimensional Schrödinger equation for the propagation of paraxial beams in strongly nonlocal nonlinear media (SNNM), the analytical solution of spatiotemporal controllable Airy-Ince-Gaussian (CAiIG) beam in elliptic coordinates is obtained. CAiIG beams are obtained by time-domain modulation of spatial Ince-Gaussian beams by Airy pulses with different initial velocities. The propagation process of CAiIG beam in SNNM is discussed by adjusting the ratio of initial incident power to critical power, initial incident velocity and ellipse parameter. The spatiotemporal CAiIG beam keeps non-dispersive characteristics in the process of propagation. According to the ratio of the initial incident power to the critical power, the beam width oscillates periodically in space (the power ratio is not equal to 1) or remains constant (the power ratio is equal to 1). The continuous conversion among spatiotemporal Airy-Ince-Gaussian beam, spatiotemporal Airy-Laguerre-Gaussian beam and spatiotemporal Airy-Hermite-Gaussian beam can be realized by adjusting the ellipse parameter. Moreover, the energy flow distributions on several propagation cross sections of CAiIG beams propagating in breathing state in SNNM are given.