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.
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