We present results of simulations aimed at tracing the formation of nuclear star clusters (NCs) and black hole (BH) seeds in the framework of the current Λcold dark matter (ΛCDM) cosmogony. These BH seeds are considered to be progenitors of the supermassive BHs that inhabit today’s galaxies. We focus on two mechanisms for the formation of BHs at high redshifts: as end-products of (1) Population III stars in metal-free haloes, and (2) runaway stellar collisions in metal-poor NCs. Our model tracks the chemical, radiative and mechanical feedback of stars on the baryonic component of the evolving haloes. This procedure allows us to evaluate when and where the conditions for BH formation are met, and to trace the emergence of BH seeds arising from the dynamical channel, in a cosmological context. BHs start to appear already at redshift ∼30 as remnants of Population III stars. The efficiency of this mechanism begins decreasing once feedbacks become increasingly important. Around redshift z∼ 15, BHs mostly form in the centre of mildly metal-enriched haloes inside dense NCs. The seed BHs that form along the two pathways have at birth a mass of around 100–1000 M⊙. The occupation fraction of BHs is a function of both halo mass and mass growth rate: at a given redshift, heavier and faster growing haloes have a higher chance to form a native BH, or to acquire an inherited BH via merging of another system. With decreasing z, the probability of finding a BH shifts towards progressively higher mass halo intervals. This is due to the fact that, at later cosmic times, low-mass systems rarely form a seed, and already formed BHs are deposited into larger mass systems due to hierarchical mergers. Our model predicts that at z= 0, all haloes above 1011 M⊙ should host a BH (in agreement with observational results), most probably inherited during their lifetime. Haloes less massive than 109 M⊙ have a higher probability to host a native BH, but their occupation fraction decreases below 10 per cent.