The primary population of extragalactic sources capable of accelerating ultra-high energy cosmic rays (UHECRs) is still unknown.
We explore two generic hypotheses for tracing the density of UHECR sources in the Universe: we assume that the UHECR production rate follows the cosmic star formation rate density or stellar mass density. For each scenario, we infer a set of constraints for the emission mechanisms in the accelerators, for their energetics, and the abundances of elements at escape from their environments by fitting together the energy spectrum and the mass composition as measured by the Pierre Auger Observatory at energy above the so-called ankle ($10^{18.7}$ eV). From these constraints, we generate sky maps above 40~EeV expected from a catalog of more than 400,000 galaxies out to 350~Mpc, which provides a near-infrared flux-limited sample to map both stellar mass and star formation rate over the full sky. Considering a scenario of intermittent sources hosted in every galaxy, we show that the main features observed in the arrival directions of UHECRs can, in turn, constrain the burst rate of the sources, also considering the magnetic effects in the cosmic web. Local magnetic fields within a few Mpc could explain why no contributions are seen from past transients in the Magellanic clouds or the Milky Way while remaining visible from more distant galaxies. Finally, we compare the burst rate inferred in a transient scenario with astrophysical candidates. The comparison shows that the best candidates are long gamma-ray bursts with low luminosities and tidal disruption events.