Certain classes of astrophysical objects produce flares that might also generate ultra-high-energy particles. If some of these particles are neutral, they could be detected in groups that would be closely correlated in both arrival direction and time. Finding such clusters in cosmic ray data would be an evidence for the existence of neutral ultra-high-energy particles, and could help determine their origins. We present the so-called stacking method to search for these space-time clustering of ultra-high-energy air showers. This method combines a time-clustering analysis with an unbinned likelihood approach. In the case of photon-induced events, background events (initiated by hadrons) can be more efficiently eliminated by using a photon tag that employs probability distribution functions to categorize each event as more likely to be either a photon or a hadron. Monte Carlo tests show that this method effectively separates photon-initiated events from hadron-initiated ones and can accurately reproduce both the number of photon events and the duration of the flare(s). With the photon tag, the stacking method can detect photon flares with only a few signal events. This approach can help identify cosmic ray sources and improve limits on the fluxes of ultra-high-energy photons.