For a normal, integral scheme $X$, for an integral scheme $Y$, for a proper, locally finitely presented morphism $f: X\to Y$, if the natural map $f^\# : \mathcal{O}_Y\to f_*\mathcal{O}_X$ is an isomorphism, then the geometric generic fiber of $f$ is integral and LCI (thus Cohen-Macaulay), but quite possibly not $R1$, i.e., not normal.

Denote by $K(Y)$ the function field of $Y$.  Denote by $X_\eta$ the fiber product of $f$ and the natural morphism $i:\text{Spec}\ K(Y) \to Y$.  Since $X$ is a normal scheme and $X_\eta$ is obtained by localization, also $X_\eta$ is a normal $K(Y)$-scheme.  Also the natural map $K(Y)\to H^0(X_\eta,\mathcal{O}_{X_\eta})$ is an isomorphism since it is the generic fiber of the isomorphism $f^\#$.  

Denote by $K(Y)^{\text{sep}}/K(Y)$ the separable closure of $K(Y)$.  Denote by $X_{\eta^s}$ the base change of $X_\eta$ to $K(Y)^{\text{sep}}$.  Since $K(Y)^{\text{sep}}/K(Y)$ is a limit of etale extensions, the same holds for $X_{\eta^s}\to X_\eta$.  In particular, $X_{\eta^s}$ is a normal scheme.  Since the base change is flat, also $K(Y)^{\text{sep}} \to H^0(X_{\eta^s},\mathcal{O}_{X_{\eta^s}})$ is still an isomorphism.  Thus, $X_{\eta^s}$ is still a normal, integral $K(Y)^{\text{sep}}$-scheme.

Finally, the field extension to the algebraic closure $\overline{K(Y)}/K(Y)^{\text{sep}}$ is purely inseparable.  Thus for the corresponding base change $X_{\overline{\eta}}$ of $X_{\eta^s}$, the morphism $X_{\overline{\eta}} \to X_{\eta^s}$ is also purely inseparable.  In particular, it is a homeomorphism (with respect to the Zariski topologies).  Since $X_{\eta^s}$ is irreducible, also $X_{\overline{\eta}}$ is irreducible.  As explained in my comments above, $X_{\overline{\eta}}$ is LCI and generically reduced, hence (everywhere) reduced.  Thus, the geometric generic fiber $X_{\overline{\eta}}$ is integral and LCI.  However, there are examples (like quasi-elliptic fibrations) where $X_{\overline{\eta}}$ is not $R1$.      

<B>Edit.</B> The OP did not ask this, but I (and my advisees) have occasionally found it useful.  In the setting above, since the divisorial part of the closed subscheme $X_{\eta}^\text{sing} \subset X_{\eta}$ (cut out by the Fitting ideal of the sheaf of relative differentials, for example) must be inseparable over $K(Y)$, this forces certain numerical invariants to be divisible by the characteristic $p$.  So if you study varieties with those numerical invariants equal to a specified (nonzero) integer, if you exclude "small characteristics" that divide that integer, then $X_{\overline{\eta}}$ will be normal.