  Supernova neutrino boosted dark matter (SN$\nu$ BDM) and its afterglow effect
have been shown to be a promising signature for beyond Standard Model (bSM)
physics. The time-evolution feature of SN$\nu$ BDM allows for %the possibly
direct inference of DM mass $m_\chi$, and results in significant background
suppression with improving sensitivity. This paper extends the earlier study
and provides a general framework for computing the SN$\nu$ BDM fluxes for a
supernova that occurs at any location in our galaxy. A bSM
$U(1)_{L_\mu-L_\tau}$ model with its gauge boson coupling to both DM and the
second and third generation of leptons is considered, which allows for both
DM-$\nu$ and DM-$e$ interactions. Detailed analysis of the temporal profile,
angular distribution, and energy spectrum of the SN$\nu$ BDM are performed.
Unique signatures in SN$\nu$ BDM allowing extraction of $m_\chi$ and detail
features that contain information of the underlying interaction type are
discussed. Expected sensitivities on the above new physics model from
Super-Kamiokande, Hyper-Kamiokande, and DUNE detections of BDM events induced
by the next galactic SN are derived and compared with the existing bounds.
