While working through a proof of this paper, at the beginning of page 42, the author seems to claim the following is true:

Let $ R\subset S$ be rings, where $ R$ is a finite type algebra over $ \mathbb F_p$ . Consider the associated map of the prime spectra $ $ \varphi:\text{Spec}(S)\rightarrow \text{Spec}(R). $ $ Suppose that $ K\subset \text{Spec}(R)$ is a constructible subset such that $ \varphi^{-1}(K)=\varnothing$ . Prove that there exists an $ R\subset R^{‘}\subset S$ , such that $ R’$ is a finite type $ R-$ algebra and that if $ $ \psi:\text{Spec}(R’)\rightarrow \text{Spec}(R). $ $ is the associated map of spectra, then $ \psi^{-1}(K)=\varnothing$ .

I believe that I have an argument for the case when $ K$ is a finite subset. One could think of $ S$ as a direct limit of its finitely generated $ R$ -subalgebras and therefore $ Spec(S)$ should equal an inverse limit of the spectra of the finitely generated $ R$ -subalgebras. For each prime in $ K$ , choose a finitely generated $ R$ -subalgebra where it does not have a preimage, and the rest is clear. However, I don’t know what to do for the general case.