$\Delta F=2$ processes in the MSSM in large $\tan\beta$ limit

We discuss corrections to $\Delta M_{B_d}$, $\Delta M_{B_s}$ and to the CP violation parameter $\epsilon$ in two examples of (generalized) minimal flavour violation models: 2HDM and MSSM in the large $\tan\beta$ limit. We show that for $H^+$ not too heavy, $\Delta M_{B_s}$ in the MSSM with heavy sparticles can be substantially smaller than in the SM due the charged Higgs box contributions and in particular due to the growing like $\tan^4\beta$ contribution of the double penguin diagrams involving neutral Higgs boson exchanges.