Suppose that a scheme $S$ is separated, excellent, and has finite Krull dimension. Which of the following statements are true:

If $S$ is regular, then it can be presented as a projective limit of smooth $\mathbb{Z}$-schemes.

If $S$ is regular, then it can be presented as a projective limit of schemes that are smooth over finite type regular $\mathbb{Z}$-ones.

Any (i.e. not necessarily regular) $S$ can be presented as a projective limit of schemes that are smooth over finite type $\mathbb{Z}$-ones.

Also, could one assume the connecting morphisms in the limits above to be affine?

I would be deeply grateful for any statements, examples, or references here! Are there any additional restrictions on $S$ needed? Is such a statement true at least locally? Are there any counter-examples known?