Suppose I have a certain (contravariant) moduli functor $M:Schemes \to Groupoids$ that is represented by a quotient stack $[X//G]$ where $X$ is a scheme and $G$ a linearly reductive group. Roughly speaking $[X//G]$ is the fine moduli space for $M$; hence it carries a universal family and there's a correspondence between families of objects of $M$ over a scheme $S$ and maps $S\to [X//G]$. Now, I claim that also $X$ carries a universal family, obtained via pull-back from $[X//G]$, and it is constant along the fibers of the quotient map $X \to [X//G]$.

If $S$ is once again a family of objects of $M$, what can one say of maps from $S\to X$? That the family over $S$ induces a map to $X$ that is unique up to $G$-action? That such a map is unique as long as $X\to [X//G]$ has a section?

Basically, by the existence of the universal family over $X$ a map $S\to X$ induces a family over $S$ but not in a unique way. So, in the other direction, given a family over $S$ this induces an orbit under $G$ of maps, but there's no canonical choice of a particular map in this orbit.