We probe the current-induced magnetic switching of insulating antiferromagnet-heavy-metal systems, by electrical spin Hall magnetoresistance measurements and direct imaging, identifying a reversal occurring by domain wall (DW) motion. We observe switching of more than one-third of the antiferromagnetic domains by the application of current pulses. Our data reveal two different magnetic switching mechanisms leading together to an efficient switching, namely, the spin-current induced effective magnetic anisotropy variation and the action of the spin torque on the DWs.