  We investigate the evolution of intermediate-mass (IMBHs), stellar (BHs) and
binary black holes (BHBs), deposited near a supermassive black hole (SMBH) by a
population of massive star clusters. Stellar BHs rapidly segregate around the
SMBH, driving the formation of extreme mass-ratio inspirals that coalesce at a
rate $\Gamma= 0.02-0.2$ yr$^{-1}$ Gpc$^{-3}$ at redshift $z=0$. A few IMBHs
orbiting the SMBH favour the formation of massive pairs that coalescence within
a Hubble time, being the merger rate for this channel $\Gamma =0.03$ yr$^{-1}$
Gpc$^{-3}$. Recoiling kicks post-merger can eject the remnant from the galaxy
centre, especially in dwarf galaxies. Our results suggest that this mechanism
can lead to up to $10^5$ ejected SMBH within 1 Gpc. An IMBH co-existing with a
few single and binary BHs in the same cluster can affect significantly their
evolution, either driving binary disruption, yielding to intermediate-mass
ratio inspirals (merger rate $\Gamma =9.5$ yr$^{-1}$ Gpc$^{-3}$), or boosting
BHBs coalescence ($\Gamma =2-8$ yr$^{-1}$ Gpc$^{-3}$). In a few simulations,
the SMBH boosts BHBs coalescence, leading this process to a merger rate $\Gamma
=1$ yr$^{-1}$ Gpc$^{-3}$. We note that BHBs experiencing a merger in a galactic
nucleus can be erroneously estimated $\sim 30\%$ heavier than it really is
because of the Doppler shift of the wave frequency as caused by the rapid
motion around the SMBH. All our simulations are carried out using an $N$-body
code tailored to treat close encounters and post-Newtonian dynamics, that
includes also the galaxy field and dynamical friction in the particles'
equation of motion.
