We present searches for ultra-high energy (UHE) neutrinos (> 0.1 EeV) with the Pierre Auger Observatory, following the detections of various types of transient astrophysical sources. These include binary black hole (BBH) mergers, detected via gravitational waves (GWs) by the LIGO and Virgo detectors. The results of the searches for a flux of UHE neutrinos from the BBH mergers combined are reported and include events from the currently ongoing third observing run of the LIGO and Virgo GW detectors. Another source of interest in the context of multi-messenger (MM) astrophysics is TXS 0506+056, a blazar that was found to emit both photons and neutrinos in a correlated manner, but at lower energies than accessible with the Pierre Auger Observatory. Follow-up searches for UHE neutrinos were performed during periods of increased emission of high-energy photons and neutrinos from TXS 0506+056 and the corresponding results are reported.