I am currently on the last chapters in Liu's book and I am trying to solve the following problem, which is the first step in showing that a certian reduction map is well-defined: Let $X \rightarrow T$ be a flat proper morphism to a regular locally Noetherian scheme $T$. Let us suppose that $T$ is irreducible with generic point $\xi$. Let $x \in X_\xi$ be a closed point and $D$ its Zariski closure in X, endowed with the reduced closed subscheme structure.
a) Show that $D$ is finite over $T$.
b) If $x$ is rational over $k(\xi)$, show that $D \rightarrow T$ is an isomorphism. One can thus define a reduction map $X_\xi(k(\xi)) \rightarrow X_t(k(t))$ for every $t \in T$.
Here is my work so far, I have only done some work on a), but doesn't seem to get all the way through. What would show that D is finite over T is that it is quasi-finite, since D is proper over T and by Zariski's main theorem proper + quasi-finite implies finite. However, here is where I get stuck. I see that $X$ is (universally) catenary, and so is T. I guess that from this, one should be able to deduce that $\dim D \cap X_t = 0$ for every $t \in T$.So, I tried to use the following identities, which follows from X catenary: $$dim X_t = \text{codim}(D \cap X_t, X_t)+ \dim(D \cap X_t)$$ and $\dim D = \text{codim}(D \cap X_t, D)+\dim(D \cap X_t).$$
I then try to use that $X \rightarrow T$ is flat, but with no luck, to show that $\dim (D \cap X_t) = 0$. Am I thinking completely wrong here? I would appreciate any hint to $a)$ or $b)$ (or if you can see no hint, an answer is OK, though I prefer a hint).
Suppose $\dim T=1$. The generic fiber of $D$ is just one point, so it is equidimensional. By the dimension formula ([Liu, 8.2.5]), we have $\dim D\le 1$. This implies that the fibers of $D\to T$ are finite, hence $D\to T$ is quasi-finite, thus finite by properness. If $x$ is a rational point, then $D\to T$ is birational and finite, hence an isomorphism because $T$ is regular.
If $\dim T\ge 2$, let $T'\to T$ be the blowing-up at a closed point of codimension $\dim T$. Suppose further that $T$ is affine. Then $T'\to T$ is projective: $T'$ can be identified with a closed subscheme of some projective space $X=\mathbb P^N_T$. Let $x$ be the generic point of $T'$. This is a rational point of $X_\xi$. But $D=T'$ and $D\to T$ is not finite.