-  Quasi-perpendicular collisionless shocks undergo structural changes with the increase of the Mach number. These changes are related to the increasing role of the reflected ions, which have a highly nongyrotropic distribution. Eventually, it is expected that the shock front becomes nonstationary. At low and moderate Mach numbers, the fraction of reflected ions is small, yet recent observations show the existence of a well-pronounced structure of the postshock magnetic field in the close vicinity of the transition layer. Large amplitude oscillations were earlier interpreted as waves generated by the shock front or passing through the shock in the downstream direction. Here we show, using two-dimensional hybrid simulations of quasi-perpendicular shocks, that the gyration of the directly transmitted ions downstream of the ramp produces the spatial pressure variations, which are accompanied with the observed magnetic oscillations due to the momentum conservation. In a wide range of the upstream ion temperatures, the low and moderate-Mach-number shocks remain stationary and one-dimensional, so that the magnetic and electric field depend only on the coordinate along the shock normal. The downstream ion distributions gradually gyrotropize due to the collisionless mixing of gyrophases. Nonstationary effects in these shocks do not affect noticeably the ion dynamics. However, we find that with the increase of the Mach number, shocks form rippled fronts in the low-β and moderate-β regimes.