Let $X$ be a separated irreducible variety which is regular in codimension $1$ (I want to talk about Weil divisors), and write $\mathbb{P}^n_X = \mathbb{P}^n_k \times X$ for the projective space over $X$. Now if $Y \subset \mathbb{P}^n_X$ is a prime Weil divisor, i.e. an irreducible closed subvariety of codimension $1$, we can consider the generic fiber $Y_K$ (here $K$ is the function field of $X$).
Since closed embeddings are stable under base change, $Y_K$ is a closed subscheme of $\mathbb{P}^n_K$, and clearly $Y_K \neq \varnothing$ if and only if the projection of $Y$ to $X$ is dense. My question: in this case ($Y_K \neq \varnothing$), can we conclude that $Y_K$ is an integral subvariety of $\mathbb{P}^n_K$ of codimension $1$? Hopefully this is true, since then we will have a well-defined degree map $\text{Pic } \mathbb{P}^n_X \to \mathbb{Z}$.
Edit: Maybe I should be asking a more general question: if $Y \to X$ is a dominant morphism of (separated, if necessary) irreducible varieties and $K$ denotes the function field of $X$, is $Y_K = Y \times_X \text{Spec } K$ an irreducible variety over $K$? The codimension $1$ assertion seems easier, since it is local, but I don't know a good criterion for irreducibility.