  We analyse scalar flavour-changing neutral currents of down-type quarks in
models with two Higgs doublets, coupled separately to up- and down-type quarks,
in the limit where the ratio of the two expectation values ($\tan \beta =
v_u/v_d$) is large. We clarify the origin of this phenomenon, both in $\Delta
F=1$ and $\Delta F=2$ processes, analysing differences and analogies between
supersymmetric and non-supersymmetric models. We confirm previous findings of a
sizeable enhancement at large $\tan\beta$ of specific $\Delta F=1$ and $\Delta
F=2$ amplitudes in the MSSM and, in these cases, we discuss how
large-$\tan\beta$ corrections can be controlled beyond lowest order. Finally,
we emphasize the unique role of the rare processes $B_{s,d} \to \tau^+ \tau^-$
and $B_{s,d} \to \mu^+ \mu^-$ in probing this scenario.
