  We analyse the rare processes $K \to \pi\nu\bar\nu$ in view of the recent
hints of violations of Lepton Flavour Universality (LFU) observed in B meson
decays. If, as suggested by present data, the new interactions responsible for
LFU violations couple mainly to the third generation of left-handed fermions,
$K \to \pi\nu\bar\nu$ decays turn out to be particularly interesting: these are
the only kaon decays with third-generation leptons (the $\tau$ neutrinos) in
the final state. In order to relate B-physics anomalies and K decays we adopt
an Effective Field Theory approach, assuming that the new interactions satisfy
an approximate $U(2)_q\times U(2)_\ell$ flavour symmetry. In this framework we
show that O(1) deviations from the Standard Model predictions in $K \to
\pi\nu\bar\nu$ branching ratios, closely correlated to similar effects in $B
\to K^{(*)}\nu\bar\nu$, are naturally expected. The correlation of
$\mathcal{B}(K \to \pi\nu\bar\nu)$, $\mathcal{B}(B \to K^{(*)}\nu\bar\nu)$, and
the LFU violations in B decays would provide a very valuable tool to shed more
light on this interesting phenomenon.
