What is a quotient of an affine scheme that is not a universal quotient? Let's recall some terminology.

Suppose that $k$ is an algebraically closed field and $G$ is a reductive group acting on an affine scheme $X$. Theorem 1.1 of Geometric Invariant Theory states that the uniform categorical quotient $X//G$ of $X$ exists.

In other words, $X \to X//G$ is universal with respect to $G$-invariant morphisms out of $X$ and this property persists under base change by a *flat* morphism $T \to X//G$.

When $\text{char}(k)=0$, the theorem states that $X \to X//G$ is a universal categorical quotient, so that the universal property persists under base change by an *arbitrary* morphism $T \to X//G$.

What is an example where $X \to X//G$ is not a universal quotient?

I'd be particularly interested in the case where the stabilizers of the action on $X$ are all linearly reductive.