  Leptoquarks (LQs) have been extensively studied in the context of $B$
anomalies. When $U(1)_{L_\mu-L_\tau}$ is introduced to a scalar LQ model with
the LQ $S_1$ charged under the new symmetry, $S_1$ primarily couples to the
third-generation leptons while its couplings to first and second-generation
leptons are naturally suppressed. Furthermore, only $S_1$ in the scalar LQ
models has the feature that down-type quarks merely couple to neutrinos but not
the charged leptons, avoiding strict restrictions from $b\to s \mu^+ \mu^-$.
With this distinctive characteristic of $S_1$, we investigate its impact on
rare processes involving the $d_i \to d_j \nu \bar\nu$ transitions. Under the
dominant constraints from $\Delta F=2$ processes, we find that the $S_1$
contributions to the branching ratios (BRs) of $B\to K(K^*) \nu \bar\nu$ and
$K_L \to \pi^0 \nu \bar\nu$ can be factorized into the same multiplicative
factor multiplying the standard model predictions. Enhancement in the BRs can
possibly exceed a factor of 2. In particular, ${\cal B}(K^+\to \pi^+ \nu
\bar\nu)$ can reach the upper $1\sigma$ error of the experimental value, i.e.,
$\simeq 15.4 \times 10^{-11}$. We also show that the model can fit the new
world averages of $R(D)$ and $R(D^*)$.
