Massive black holes (BHs) inhabit local galaxies, including the Milky Way and some dwarf galaxies. BH formation, occurring at early cosmic times, must account for the properties of BHs in today's galaxies, notably why some galaxies host a BH, and others do not. We investigate the formation, distribution and growth of BH `seeds' by using the adaptive mesh refinement code Ramses. We develop an implementation of BH formation in dense, low-metallicity environments, as advocated by models invoking the collapse of the first generation of stars, or of dense nuclear star clusters. The seed masses are computed one-by-one on-the-fly, based on the star formation rate and the stellar initial mass function. This self-consistent method to seed BHs allows us to study the distribution of BHs in a cosmological context and their evolution over cosmic time. We find that all high-mass galaxies tend to a host a BH, whereas low-mass counterparts have a lower probability of hosting a BH. After the end of the epoch of BH formation, this probability is modulated by the growth of the galaxy. The simulated BHs connect to low-redshift observational samples, and span a similar range in accretion properties as Lyman-Break Analogs. The growth of BHs in low-mass galaxies is stunted by strong supernova feedback. The properties of BHs in dwarf galaxies thus remain a testbed for BH formation. Simulations with strong supernova feedback, which is able to quench BH accretion in shallow potential wells, produce galaxies and BHs in better agreement with observational constraints.

16 pages, 16 figures, matched the published version