Let $k$ be a field of characteristic not 2 or 3. Then the set of elliptic curves over $k$ can be parametrized by the affine variety $S=D(4a^3+27b^2)\subset\mathbb{A}^2_k$ via the family $E\to S$ where $E\subset\mathbb{P}^2_k\times S$ given is by the equation $y^2=x^3+ax+b$.

Is this in some mild sense universal? For example is every family of elliptic curves $E'\to S'$ over some smooth $k$-variety $S'$ the pull-back of $E\to S$ via a (necessarily non-unique) morphism $S'\to S$? If not, at least locally?

Are there analogous constructions for abelian varieties of higher dimension? Is there a family of abelian varieties of dimension $g$ over some irreducible (affine? smooth?) variety $S$ over $k$ such that every abelian variety of dimension $g$ over $k$ is a fiber over a rational point of $S$?

In the literature I have only found (stronger) results over algebraically closed fields or when we additionally require something like a $\Gamma_1(N)$ structure or something.