Let $ f:X\rightarrow S$ be a proper morphism of schemes with Noetherian target. The theorem on formal functions says that for any point $ s\in S$ there is an isomorphism between inverse limits of $ (f_*O_X)_s/\mathfrak{m}_s^n (f_*O_X)_s$ and $ \Gamma(X_s, O_X\otimes_{O_S}O_S/\mathfrak{m}_s^n O_S)$ , if I understand correctly. If $ f$ happens to be flat and cohomologically flat in degree 0, then we know the isomorphism between inverse limits is actually an isomorphism at $ n$ -th stage for any $ n>0$ .

- Is there an example of a morphism $ f$ such that the induced comparison morphism is an isomorphism at $ n$ -th stage (and thus isomorphism at $ m$ -th stage for every $ 0<m<n$ ) and is not an isomorphism at any later stage?
- Is there a description of $ (f_*O_X)_s$ in terms of fiberwise information?