  We propose the local $(B-L)_{\mu\tau}$ model, which minimally retains the
local $B-L$ extension for the sake of neutrino phenomenologies, and at the same
time presents an invisible gauge boson $Z'$ with mass $\sim {\cal O}(10)$ MeV
to account for the discrepancy of the muon anomalous magnetic moment. However
such a scenario is challenged by flavor physics. To accommodate the correct
pattern of Cabibbo-Kobayashi-Maskawa matrix, we have to introduce either a
$SU(2)_L$ doublet flavon or vector-like quarks plus a singlet flavon. In either
case $Z'$ induces flavor changing neutral current (FCNC) in the quark sector at
tree-level. We find that the former scheme cannot naturally suppress the FCNC
from the down-type quark sector and thus requires a large fine-tuning to avoid
the stringent $K \to \pi \nu\bar \nu$ bound. Whereas the latter scheme, in
which FCNC merely arises in the up-type quark sector, is still free of strong
constraint. In particular, it opens a new window to test our scenario by
searching for flavor-changing top quark decay mode $t \rightarrow
u/c+$(invisible), and the typical branching ratio $\sim\mathcal{O}(10^{-4})$.
