# Support of a coherent sheaf over a fiber product scheme

I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:

Let $$X,S$$ be two $$K$$-scheme of finite type with $$K$$ an algebraically closed field. Let $$\mathcal{E}$$ be a coherent sheaf over $$X\times_{Spec(K)} S=X\times S$$ flat over $$S$$. Then, $$Supp(\mathcal{E})\subset X\times S$$ is proper over $$S$$

I know this is obviously false when we take the fibre product of $$X$$ and $$S$$ over another scheme $$Y$$, say. I'm wondering if in this particular case it is true that $$\pi_S:X\times S\to S$$ is of finite type (perhaps by using that being of finite type is preserved under base change) so that we can use the following lemma https://stacks.math.columbia.edu/tag/0CYL from StackExchange.

If the above fact it's not true, could you please give me a counterexample?

Take $$X = S = \mathbb{A}^1$$, $$Z = \{xy = 1\} \subset X \times S$$ (where $$x$$ is the coordinate on $$X$$ and $$y$$ on $$S$$), and let $$\mathcal{E}$$ be the structure sheaf of $$Z$$.