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Context. Low-luminosity supermassive and stellar-mass black holes (BHs) may be embedded in a collisionless and highly magnetized plasma. They show nonthermal flares indicative of particles being accelerated up to relativistic speeds by dissipative processes in the vicinity of the BH. During near-infrared flares from the supermassive BH Sagittarius A* (Sgr A*), the GRAVITY Collaboration detected circular motion and polarization evolution, which suggest the presence of transient synchrotron-emitting hot spots moving around the BH. Aims. We study 3D reconnecting current layers in the magnetosphere of spinning BHs to determine whether plasma-loaded flux ropes which are formed near the event horizon could reproduce the hot spot observations and help constrain the BH spin. Methods. We performed global 3D particle-in-cell simulations in Kerr spacetime of a pair plasma embedded in a strong and large-scale magnetic field originating in a perfectly conducting disk in prograde Keplerian rotation. Results. A cone-shaped current layer develops which surrounds the twisted open magnetic field lines threading the event horizon. Spinning magnetic field lines coupling the disk to the BH inflate and reconnect a few gravitational radii above the disk. This quasi-periodic cycle accelerates particles, which accumulate in a few macroscopic flux ropes rotating with the outermost coupling magnetic field line. Once flux ropes detach, they propagate in the current layer following what appears as a rapidly opening spiral when seen face-on. A single flux rope carries enough relativistic electrons and positrons to emit synchrotron radiation at levels suitable to reproduce the peak-luminosity of the flares of Sgr A* but it quickly fades away as it flows away. Conclusions. Our kinematic analysis of the flux ropes motion favors a BH spin of 0.65 to 0.8 for Sgr A*. The duration of the flares of Sgr A* can only be explained provided the underlying magnetic loop seeded in the disk mid-plane has a finite lifetime and azimuthal extension. In this scenario, the hot spot corresponds to a spinning arc along which multiple reconnection sites power the net emission as flux ropes episodically detach. [ABSTRACT FROM AUTHOR]