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We discuss the conditions for the existence of extended matter configurations orbiting in the ergoregion or close to the outer ergosurface of the Kerr black hole ("dragged" configurations). The co-rotating tori under consideration are perfect fluid configurations with barotropic equations of state, orbiting on the equatorial plane of the central Kerr black hole. The possibility of magnetized tori with a toroidal magnetic field is also discussed. Indications on the attractors where dragged tori can be observed are provided with the analysis of the fluid characteristics and geometrical features, relevant for the torus stability and their observations. QPO emissions from the inner edges of the dragged tori are also discussed. We argue that the smaller dragged tori could be subjected to a characteristic instability: the effect of the frame-dragging. This possibility is thoroughly explored. This can finally lead to the destruction of the torus (disk exfoliation) which can combine with accretion and processes present in the region very close to the black hole horizon. Tori are characterized according to the central attractor dimensionless spin. These structures can be observed in orbiting black holes with dimensionless spin a > 0.9897 M.