The context of this question is my attempt to generalise the Hurwitz theorem to varieties that aren't necessarily curves. In the case of curves, we have that a closed point always maps to a closed point because we are generally only dealing with finite morphisms. So we get a morphism from a prime divisor to a prime divisor, which we can extend linearly with ramification indices.
But as soon as we leave the world of curves this seems more difficult. The situation I am in is if we have a normal variety $Y$ with a projective birational morphism from a smooth variety $\pi: X \rightarrow Y$. Now if $p \in X$ is the generic point of a codimension $1$ subset, then why should it be true that $f(p)$ also has codimension $1$?
My attempt so far is to notice that since the morphism is dominant we have an injective morphism on local rings $\mathcal{O}_{Y, f(p)} \rightarrow \mathcal{O}_{X, p}$ from which we get that $\dim \mathcal{O}_{Y, f(p)} \leq \dim \mathcal{O}_{X, p}$. But to obtain equality we would normally want the extension of local rings to be integral. Is there any way to deduce integrality from the fact that the morphism $\pi$ is proper? If it were affine I think we could, because we would have a univerally closed morphism of affine schemes. But in this case it doesn't seem so obvious. Is it even true?
This is false for any dimension bigger than 1: take $f:X \rightarrow Y$ to be the blowup of a smooth point, and $p$ the generic point of the exceptional divisor.
On the other hand, you say you are interested in the situation when $f: X \rightarrow Y$ is a birational morphism to a normal variety. Understanding the relationship between the canonical divisors of $X$ and $Y$ in this case is a fundamental issue in minimal model theory. Under suitable assumptions, there is a formula
$$K_X = f^* K_Y + \sum_i a_i E_i$$ where the sum is over all prime divisors on $X$ contracted by $f$, and the coefficients $a_i$ is a rational number called the discrepancy of $E_i$. The values of the $a_i$ depend on the singularities of $Y$, and imposing various bounds on the $a_i$ defines important classes of singularities such as terminal, canonical, and log canonical.
More details can be found in various texts such as those of Koll'ar--Mori, Matsuki, Debarre...