  We investigate the formation and growth of massive black hole (BH) seeds in
dusty star-forming galaxies, relying and extending the framework proposed by
Boco et al. 2020. Specifically, the latter envisages the migration of stellar
compact remnants (neutron stars and stellar-mass black holes) via gaseous
dynamical friction towards the galaxy nuclear region, and their subsequent
merging to grow a massive central BH seed. In this paper we add two relevant
ingredients: (i) we include primordial BHs, that could constitute a fraction
$f_{\rm pBH}$ of the dark matter, as an additional component participating in
the seed growth; (ii) we predict the stochastic gravitational wave background
originated during the seed growth, both from stellar compact remnant and from
primordial BH mergers. We find that the latter events contribute most to the
initial growth of the central seed during a timescale of $10^6-10^7\,\rm yr$,
before stellar compact remnant mergers and gas accretion take over. In
addition, if the fraction of primordial BHs $f_{\rm pBH}$ is large enough,
gravitational waves emitted by their mergers in the nuclear galactic regions
could be detected by future interferometers like Einsten Telescope, DECIGO and
LISA. As for the associated stochastic gravitational wave background, we
predict that it extends over the wide frequency band $10^{-6}\lesssim f [{\rm
Hz}]\lesssim 10$, which is very different from the typical range originated by
mergers of isolated binary compact objects. On the one hand, the detection of
such a background could be a smoking gun to test the proposed seed growth
mechanism; on the other hand, it constitutes a relevant contaminant from
astrophysical sources to be characterized and subtracted, in the challenging
search for a primordial background of cosmological origin.
