  We revisit the phases of the pairing and sinking of BHs in galaxy mergers and
circunmunclear disks in light of the results of recent simulations with massive
BHs embedded in predominantly gaseous backgrounds. After a general overview we
discuss the importance of a fast orbital decay regime dominated by global disk
torques rather than by the local dynamical friction wake. This regime can
dominate at BH binary separations of a few tens of parsecs and below, following
a phase of orbital circularization dominated by local dynamical friction. It is
similar to Type-I migration in planetary evolution. It can bring the black
holes to separations small enough for gravitational waves to take over on a
timescale ranging from less than $\sim 10^7$ yr to up to $10^8$ yr, depending
on whether the interstellar medium is smooth or clumpy. Eventual gap opening at
sub-pc scale separations slows down but does not interrupt the orbital
decay.Subsequently, we discuss a new intriguing connection between the
conditions required for rapid orbital decay of massive BH binaries and those
required for prominent gas inflows in gas-rich galaxies. We derive a condition
for the maximum inflow rate that a circumnuclear disk can host while still
maintaining a sufficiently high gas density at large radii to sustain the decay
of a BH binary. We find that gas inflows rates exceeding 10 $M_{\odot}$/yr,
postulated to form massive BH seeds in some direct collapse models, would
stifle the sinking of massive BH binaries in gas-dominated galactic nuclei.
Vice-versa, lower inflow rates, below a solar mass per year, as required to
feed typical AGNs, are compatible with a fast orbital decay of BH binaries
across a wide range of masses.
