  Proto-neutron stars formed during core-collapse supernovae are hot and dense
environments that contain a sizable population of muons. If these interact with
new long-lived particles with masses up to roughly 100 MeV, the latter can be
produced and escape from the stellar plasma, causing an excessive energy loss
constrained by observations of SN 1987A. In this article we calculate the
emission of light dark fermions that are coupled to leptons via a new massive
vector boson, and determine the resulting constraints on the general parameter
space. We apply these limits to the gauged $L_\mu-L_\tau$ model with dark
fermions, and show that the SN 1987A constraints exclude a significant portion
of the parameter space targeted by future experiments. We also extend our
analysis to generic effective four-fermion operators that couple dark fermions
to muons, electrons, or neutrinos. We find that SN 1987A cooling probes a
new-physics scale up to $\sim7$ TeV, which is an order of magnitude larger than
current bounds from laboratory experiments.
