$DeclareMathOperatorel{ell}$In the topological case, given a group $G$, we can define the classifying space $BG$ for principal $G$-bundles and there is a universal $G$-principal bundle $EG to BG$ classified by the identity morphism $BG to BG$ such that any other principal bundle over $X$ is given by a pullback of the universal bundle along some map $X to BG$. Is there an analog construction for algebraic stacks?

That is, take some algebraic stack classifying a family of objects over a given scheme, e.g. the moduli stack of elliptic curves. Is there any way to make sense of “the universal elliptic curve classified by the identity morphism $mathcal{M}_{el} to mathcal{M}_{el}$“? My problem is I’m not sure how to evaluate an algebraic stack on an algebraic stack – we can glue the value of a stack along a cover of a scheme, but is there any reason to expect a similar property for covers of algebraic stacks? Will the resulted object be an algebraic stack?

I’ve found this “universal bundle” mentioned in this nLab article, where they even take its bundle of differential forms, but I couldn’t find any reference or explanation, so any reference would be very welcome.