  The latest measurements of rare $b\to s$ decays in the LHCb experiment have
led to results in tension with the predictions of the standard model (SM),
including a tentative indication of the violation of lepton flavor
universality. Assuming that this situation will persist because of new physics,
we explore some of the potential consequences in the context of the SM extended
with the seesaw mechanism involving right-handed neutrinos plus effective
dimension-six lepton-quark operators under the framework of minimal flavor
violation. We focus on a couple of such operators which can accommodate the
LHCb anomalies and conform to the minimal flavor violation hypothesis in both
their lepton and quark parts. We examine specifically the
lepton-flavor-violating decays $B\to K^{(*)}\ell\ell'$, $B_s\to\phi\ell\ell'$,
$B\to(\pi,\rho)\ell\ell'$, and $B_{d,s}\to\ell\ell'$, as well as $K_L\to e\mu$
and $K\to\pi e\mu$, induced by such operators. The estimated branching
fractions of some of these decay modes with $\mu\tau$ in the final states are
allowed by the pertinent experimental constraints to reach a few times
$10^{-7}$ if other operators do not yield competitive effects. We also look at
the implications for $B\to K^{(*)}\nu\nu$ and $K\to\pi\nu\nu$, finding that
their rates can be a few times larger than their SM values. These results are
testable in future experiments.
