I can prove a certain statement for any scheme that can be presented as the limit of an essentially affine (filtering) projective system of smooth varieties over a perfect field such the connecting morphisms are dominant. In this text I only treat schemes that are excellent separated of finite Krull dimension. So, I have the following questions.

Is there a shorter description of schemes that can be presented as limits of this sort (either of all ones, or of excellent separated of finite Krull dimension)?

Is there an interesting subclass in the class of all limit schemes of this sort? I don't want to restrict myself to affine schemes.

If no nice answers to questions 1 and 2 will be given, I will need a term for limits of this sort. Any suggestions?:)