  We analyse various flavour changing processes like $t\to hu,hc$, $h\to \tau
e,\tau\mu$ as well as hadronic decays $h\to bs,bd$, in the framework of a class
of two Higgs doublet models where there are flavour changing neutral scalar
currents at tree level. These models have the remarkable feature of having
these flavour-violating couplings entirely determined by the CKM and PMNS
matrices as well as $\tan\beta$. The flavour structure of these scalar currents
results from a symmetry of the Lagrangian and therefore it is natural and
stable under the renormalization group. We show that in some of the models the
rates of the above flavour changing processes can reach the discovery level at
the LHC at 13 TeV even taking into account the stringent bounds on low energy
processes, in particular $\mu\to e\gamma$.
