The semicontinuity theorem (Hartshorne III.11) states that the ranks of cohomology groups on the fibers of a morphism is a semicontinuous function. More precisely, given a projective morphism $f:X\to Y$ of noetherian schemes and a coherent sheaf $F$ on $X$, flat over Y, then the function
$$
h^i(y,F)=\dim_{k(y)}H^i(X_y,F_y)
$$is upper semicontinuous as a function of $y$. Here $X_y$ denotes the fibre of $f$ over $y$. This is used widely in algebraic geometry.
Let $X$ and $Y$ be complex manifolds and $f:X \to Y$ a holomorphic map.
If $f$ is surjective then, by Sard's theorem, the generic fiber of $f$ has dimension $\dim X - \dim Y$. So, once we prove upper semicontinuity, we will know that all fibers have dimension at least $\dim X - \dim Y$.
To show semicontinuity, it is enough to show $\{ x : \dim f^{-1}(f(x))=0 \}$ is open. Once we have done this, suppose that $x$ lies on a $b$-dimensional component of $f^{-1}(f(x))$. Then we can choose a neighborhood $U$ of $x$ and a map $g: U \to \mathbb{C}^b$ such that the fiber of $f \times g : U \to Y \times \mathbb{C}^b$ though $x$ is just the singleton $\{ x \}$. There will then be some open neighborhood $V$ of $x$ where the fibers of $f \times g$ are finite, and hence the fibers of $f$ have dimension $\leq b$.
So, suppose that $f^{-1}(f(x))$ is finite. We must build an open neighborhood of $x$ where all fibers are finite. Shrinking $X$ around $x$, we may assume that $f^{-1}(f(x)) = \{ x \}$ and that $X$ is open in $\mathbb{C}^{d}$ for $d = \dim X$. Let $B$ be a closed ball in $X$ around $x$. Let $B^{\circ}$ be the interior and $\partial B$ the boundary. Write $y = f(x)$.
Now, $\partial B$ is compact, so $f(\partial B)$ is closed in $Y$. Since $f^{-1}(y) = \{ x \}$, we see that $y \not \in f(\partial B)$, so we can take an open set $V$ around $y$ disjoint from $f(\partial B)$. Take $U = f^{-1}(V) \cap B^{\circ}$. I claim that, for any $x' \in U$, the fiber $f^{-1}(f(x')) \cap U$ is finite.
Let $y' = f(x')$. Note that $y' \in V$ and hence $f^{-1}(y') \cap U = f^{-1}(y') \cap B^{\circ} = f^{-1}(y') \cap B$. The last is a closed subset of the compact set $B$, hence is compact. So far, we have not used complex geometry in any way.
Let $z$ be any holomorphic function on $U$. Then the restriction of $z$ to $f^{-1}(y') \cap U$ is a holomorphic function on a compact complex manifold, so it is locally constant by the maximum modulus principle. (I am glossing over the fact that $f^{-1}(y') \cap U$ could have singularities.) Applying this fact with $z_1$, $z_2$, ..., $z_d$ the coordinate functions on $U \subset \mathbb{C}^d$, we see that $f^{-1}(y') \cap U$ is finite.
Best Answer
Theorem (EGA IV 13.1.3): Let $f \colon X \to Y$ be a morphism of schemes, locally of finite type. Then $$x \mapsto \dim_x(X_{f(x)})$$ is upper semi-continuous.
Corollary (Chevalley's upper semi-continuous theorem, EGA IV 13.1.5): Let $f \colon X \to Y$ be proper, then: $$y \mapsto \dim(X_y)$$ is upper semi-continuous.
Corollary (SGA3, ??): Let $X/Y$ be a group scheme, locally of finite type. Then $$y \mapsto \dim(X_y)$$ is upper semi-continuous.
Proof: The dimension of a group scheme over a field is the same as the dimension at the identity. Thus the function $$y \mapsto \dim(X_y)$$ is the composition of the continuous function $y \to e(y)$ and the upper semi-continuous function $x \mapsto \dim_x(X_{f(x)})$.
Concerning your application: The fiber dimensions of the stabilizer group scheme Stab/X is upper semi-continuous, but the "diagonal" $G \times X \to X \times X$ does not always have this property (unless it is proper, i.e., "$G$ acts properly").