If $f:X\rightarrow Y$ is a flat morphism of schemes then the map $\Gamma(Y,\mathcal{O}_Y)\rightarrow\Gamma(X,\mathcal{O}_X) $ is flat.
This was used in the proof of proposition 11.48 in the book of Gortz -Wedhorn to prove that one can pullback Cartier divisors under a flat morphism. I have been able to understand the proposition without using this (and using the right definition of the function field) but I still don't find a proof or counterexample for it.
Does someone knows if it is true?
It would be interesting to discuse the case in which $f$ is an open immersion as well. This correspond to the restriction morphisms of the structure sheaf being flat.
Also, of course this is true if both $X$ and $Y$ are affine.