Black holes (BH) acquire relativistic magnetospheres by accreting magnetized gas. Once they collect significant magnetic flux across the horizon, aided by the spin they can drive powerful relativistic jets by the Blandford-Znajek mechanism. Large enough BH magnetic flux backreacts on the accretion flow, which has been described in terms of arresting or choking. Magnetic flux eruptions have been identified as the mechanism of BH magnetic flux saturation. These eruptions can potentially dissipate a large fraction of magnetic energy in the BH magnetosphere by means of relativistic magnetic reconnection, accelerating particles and producing flares of non-thermal radiation. We analyze the results of 3D general-relativistic ideal magnetohydrodynamic (GRMHD) numerical simulations of accretion flows onto magnetically saturated Kerr BHs, focusing on the initiation of magnetic flux eruptions.