  We investigate the consequences of deviations from the Standard Model
observed in $b\to s\mu\mu$ transitions for flavour-changing neutral-current
processes involving down-type quarks and neutrinos, under generic assumptions
concerning the structure of New Physics. We derive the relevant Wilson
coefficients within an effective field theory approach respecting the SM gauge
symmetry, including right-handed currents and assuming a flavour structure
based on approximate $U(2)$ symmetry, and only SM-like light neutrinos. We
discuss correlations among $B \to K^{(*)} \nu \bar \nu$ and $K\to \pi \nu \bar
\nu$ branching ratios in the case of linear Minimal Flavour Violation and in a
more general framework, highlighting in each case the role played by various
New Physics scenarios proposed to explain $b\to s\mu\mu$ deviations. This talk
is based on arXiv:2005.03734.
