I always trusted the following quite vague statement:

If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say varying in disc, then the intersection number $D_1(t)\cdots D_k(t)$ can only increase under specialisation of $t$.

Without any flatness assumption on $D_i(t)$ and $X_t$ this is false, as shown in the post Semicontinuity of degree of fibers for a proper map

I guess, under an appropiate fltaness assumption, my belief follows from the standard semi-continuity theorem (Hartshorne III.11). However, I was not able to find any precise statement/reference.

I would like to have a precise statment about the semi-continuity of intersection numbers, with possibly a reference in the literature.

This should also fit with the question More upper/lower semi-continuous functions in (algebraic) geometry?