In this paper, we explore the gravitomagnetic interaction of a black hole (BH) with a misaligned accretion disc to study BH spin precession and alignment jointly with BH mass MBH and spin parameter a evolution, under the assumption that the disc is continually fed, in its outer region, by matter with angular momentum fixed on a given direction . We develop an iterative scheme based on the adiabatic approximation to study the BH-disc co-evolution: in this approach, the accretion disc transits through a sequence of quasi-steady warped states (Bardeen-Petterson effect) and interacts with the BH until the spin JBH aligns with . For a BH aligning with a corotating disc, the fractional increase in mass is typically less than a few per cent, while the spin modulus can increase up to a few tens of per cent. The alignment time-scale is of ~105-106 yr for a maximally rotating BH accreting at the Eddington rate. BH-disc alignment from an initially counter-rotating disc tends to be more efficient compared to the specular corotating case due to the asymmetry seeded in the Kerr metric: counter-rotating matter carries a larger and opposite angular momentum when crossing the innermost stable orbit, so that the spin modulus decreases faster and so the relative inclination angle.

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