Abstract The arrival directions of ultrahigh-energy cosmic rays (UHECRs) observed above 4 × 1019 eV provide evidence of localized excesses that are key to identifying their sources. We leverage the 3D matter distribution from optical and infrared surveys as a density model of UHECR sources, which are considered to be transient. Agreement of the sky model with UHECR data imposes constraints on both the emission rate per unit matter and the time spread induced by encountered turbulent magnetic fields. Based on radio measurements of cosmic magnetism, we identify the Local Sheet as the magnetized structure responsible for the kiloyear duration of UHECR bursts for an observer on Earth and find that the turbulence amplitude must be within 0.5–20 nG for a coherence length of 10 kpc. At the same time, the burst-rate density must be above 50 Gpc−3 yr−1 for Local Sheet galaxies to reproduce the UHECR excesses and below 5000 Gpc−3 yr−1 (30,000 Gpc−3 yr−1) for the Milky Way (Local Group galaxies) not to outshine other galaxies. For the transient emissions of protons and nuclei to match the energy spectra of UHECRs, the kinetic energy of the outflows responsible for UHECR acceleration must be below 4 × 1054 erg and above 5 × 1050 erg (2 × 1049 erg) if we consider the Milky Way (or not). The only stellar-sized transients that satisfy both Hillas’ and our criteria are long-duration gamma-ray bursts.