For certain matters the henselization $R$ of $\mathbb{Q}[t]$ at $0$ is a 'reasonable approximation' for $\mathbb{Q}[[t]]$ (Artin's approximation theorem and so on). Now, I would like to study certain (\'etale) cohomological properties of a finite type $\mathbb{Q}[[t]]$-scheme $S$; I would like to say that there exists an $R$-scheme $S'$ that is 'very similar to $S$' ('from the cohomological point of view'). What are the possible methods for doing so? I guess that I should embedd $S$ into a 'family' and then apply either Artin's approximation (so, this a sort of deformation), or smooth base change, or both. Yet I would certainly be deeply grateful for any hints (and references); in particular, where can I find an argument for presenting $S$ as a 'member of a family'?

I can certainly say more on these cohomological issues; yet they are rather specific. They are somewhat related (yet more complicated) to the weight-monodromy conjecture; cf. http://www.jstor.org/stable/40067932

Upd. As we discussed in my previous question, $\mathbb{Q}[[t]]$ is the limit of its finitely generated subalgebras. Yet the corresponding etale pullback functors are not very nice, since $\mathbb{Q}[[t]]$ is not 'pro-smooth' over these subalgebras. So I suspect that 'approximating' $\mathbb{Q}[[t]]$ by $R$ is more convenient for my purposes.