Let $p\colon Z\to X$ be a morphism between irreducible varieties (= reduced schemes of finite type over $\mathbb{C}$). Assume that every fiber of $p$ over a closed point of $X$ is also irreducible. Let $X'$ be an irreducible subvariety of $X$. My question is: under which circumstances can we say that the inverse image $p^{-1}(X')$ is also irreducible? Can someone give an example of a situation where it is not irreducible?
I was thinking about a solution like this which is likely to be flawed because I am lacking experience: let $\eta$ be the generic point of $X'$. Since all fibers of $p$ over closed points are irreducibel we can conclude that $p^{-1}(\eta)$ is also irreducible. By this question the irreducible components of $p^{-1}(\eta)$ and $p^{-1}(X')$ are in one to one correspondence. Since $p^{-1}(\eta)$ is irreducible it follows that $p^{-1}(X')$ is also irreducible.
I think that this solution is flawed because I am using arguments which I cannot justify rigorously, for example I do not know if the fiber over $\eta$ is irreducible. (Note that this point will usually not be closed.)
The precise situation I have in mind is the following. It might be superfluous to say this but I want to be sure... Let $G$ be a simple, simply connected and connected linear algebraic group. Let $P$ be a (maximal) parabolic subgroup. Let $Z=G$ and $X=G/P$ and $p$ the natural projection. Let $X'$ be a Schubert variety in $X$. The fibers of $p$ over a closed (or any?) point is isomorphic to $P$, hence irreducible. In this specific situation it might be even possible to argue using that $p$ is open or closed? (I think $p$ is open but not closed but I am quite unsure. If it is closed we can conclude directly as far as I see.)
If someone can say a few words which yield to clarification I would be very happy. Thank you!
An obvious place this fails is when $p$ is the blowing up of say, a smooth point in $X$ ($\dim X\geq 2$). Then all fibers of closed points are either a point or a projective space. But the inverse image of any irreducible curve through the blown up point is not irreducible.