Transient and compact stars, like the proto-neutron star formed during a core-collapse supernova or after the merging of two neutron stars, are efficient factories of axions. In the small-coupling regime, these axions can escape freely from the source and convert into photons in the magnetic field outside the stars. We use the non-detection of gamma rays from SN 1987A to impose constraints on the axion-photon and the axion-proton couplings, and we examine the projected reach on these couplings with the observation of future astrophysical events of this kind. We study the entire pipeline (emission, conversion and detection) in an analytic, parametric way. We include a new ingredient that had not previously been considered in the context of neutron-star mergers: the material ejected after the merging, which is dense enough to suppress the axion-photon conversion.