(everything below is defined over an algebraically closed field)

Let $D$ be a (smooth) surface, and let $X \subset T \times D$ be a flat family of curves on $D$, where $T$ is irreducible. Let $E$ be a curve on $D$. Fix some $t_0 \in T$ such that $X_{t_0}$ does not contain $E$, the intersection $X_{t_0} \times_D E$ is some not necessarily reduced closed subscheme of $D$ of dimension 0.

Is it true then that there exists an open dense subset $U \subset T$, such that $X_t \times_D E$ are all isomorphic for different $t \in U$? Does one actually need to assume $D$ to be projective? smooth? Is the previous statement true for an affine $D$?