Transient generation of ultra-high-energy cosmic rays (UHECRs) has been motivated from promising candidates of UHECR sources such as gamma-ray bursts, flares of active galactic nuclei, and newly born neutron stars and magnetars. Here we propose a strategy to unveil transient sources of UHECRs from UHECR experiments. We demonstrate that the rate of UHECR bursts and/or flares is related to the apparent number density of UHECR sources, which is the number density estimated on the assumption of steady sources, and the time-profile spread of the bursts produced by cosmic magnetic fields. The apparent number density strongly depends on UHECR energies under a given rate of the bursts, which becomes observational evidence of transient sources. It is saturated at the number density of host galaxies of UHECR sources. We also derive constraints on the UHECR burst rate and/or energy budget of UHECRs per source as a function of the apparent source number density by using models of cosmic magnetic fields. In order to obtain a precise constraint of the UHECR burst rate, high event statistics above ∼ 1020 eV for evaluating the apparent source number density at the highest energies and better knowledge on cosmic magnetic fields by future observations and/or simulations to better estimate the time-profile spread of UHECR bursts are required. The estimated rate allows us to constrain transient UHECR sources by being compared with the occurrence rates of known energetic transient phenomena.