  In this work we study possible connections between $B$-meson anomalies and
Kaon physics observables in the context of combined solutions with the singlet
and triplet scalar leptoquarks $S_1$ and $S_3$. By assuming a flavor structure
for the leptoquark couplings dictated by a minimally broken $U(2)^5$ flavor
symmetry we can make a sharp connection between these two classes of
observables. We find that the bound on $\mathcal{B}(K^+ \rightarrow \pi^+
\nu\nu)$ from NA62 puts already some tension in the model, while the present
limits on $\mathcal{B}(K_L \rightarrow \mu^+ \mu^-)$ and $\mu \to e$ conversion
in nuclei can be saturated. Relaxing instead the flavor assumption we study
what values for $\mathcal{B}(K^+ \rightarrow\pi^+ \nu\nu)$, as well as for
$\mathcal{B}(K_L \rightarrow\pi^0 \nu\nu)$ and $\mathcal{B}(K_{L,S}
\rightarrow\mu^+ \mu^-)$, are viable compatibly with all other phenomenological
constraints.
