What is the relative version of a reduced scheme? I have often heard it said that it is important to think of properties of a scheme $X$ as really a special case of a property of morphisms applied to the morphism $X\to \operatorname{Spec} \Bbb Z$. 
Some examples of this pattern are being affine, separated, quasi-compact, quasi-separated and so on.
Is there a similar relative version of the notion of a scheme being reduced (or integral or irreducible)? I have searched a little bit and there seems to be no mention of such things anywhere. 
If it is true that we don't have a useful notion of such things, is there any reason why? This is of course a little vague and I am not really sure what answer would satisfy me.
Edit: Specifically for reduced versions of morphisms, I think they should have the following properties:


*

*They should be affine.

*They should be closed under base change(pullback/fiber products) and composition.

*(More speculative) The property should be expressible purely as a property about the induced morphisms on the stalks.


Reduced morphisms are defined in EGA IV.6.8.1. Irreducible morphisms are defined in EGA IV.4.5.5.
