Let $X$ be a smooth variety over a field $k$ and I'd like to think about $CH_0(X)$ the 0-Chow group i.e. the group of rational equivalent classes of 0-cycles. I'm wondering if there is any reasonable formulation to make sense of "family"/"moduli" of rational equivalent classes of 0-cycles, forming an fppf sheaf over $\text{Spec} k$? If there is such a sheaf, is there any chance the sheaf is actually representable?

A quick literature search shows some work on certain formulations of "Chow schemes". However, they seem to parametrize the 0-cycles, instead of the rational equivalent classes. But this seems to suggest that now we only need to construct a subsheaf of principal 0-cycles and then we can consider the quotient sheaf.

A known example: if X is in addition one dimensional and proper, we can speak of the Picard scheme which represents the relative Picard functor. I would really like to know what happens when the dimension goes higher.