Dimension of flat morphisms/schemes

[mnieper]

Classically, we know that a flat morphism between varieties has a constant fibre dimension. Can we use this somehow to be able to define the notion of a flat scheme of dimension $n$ in SAG? A flat morphism of (relative) dimension $n$ would then be a morphism whose fibres are flat of dimension $n$. The notion should correspond to the notion of dimension we have for smooth schemes. A finite flat scheme should be of dimension $0$.