Geometric intuition - flatness of morphism of stacks

Question

I have seen interesting geometric intuition answers about flatness, for example here and here.

I am not sure whether it still applies for morphisms of stack, and if yes how? In partitcular,

• Can we say something about equidimensinal fibers of a flat morphism of stacks?

• Can we say something about the "mass" of fibers in a flat morphism of stacks $f: M \to \Bbb A^1$? (the fibers seen as groupoids in the moduli space $M$ have a cardinality called mass)

Thank you!