  We analyze the impact of Higgs-mediated amplitudes on the rare decays KL ->
pi0 nu nu-bar and K+ -> pi+ nu nu-bar in the MSSM with large tan(beta) and
general flavour mixing. We point out that, going beyond the minimal flavour
violation hypothesis, Z-penguin amplitudes generated by charged-Higgs exchange
can induce sizable modifications of K -> pi nu nu-bar rates. Interestingly,
these effects scale as tan^4(beta) at the amplitude level. For large values of
tan(beta), this mechanism allows deviations from the SM expectations even for
tiny (CKM-type) off-diagonal mixing terms in the right-handed squark sector.
