As far as I understand it, in recent years there has been a lot of progress on generalizations of classical deformation theory in characteristic 0 using tools such as simplicial deformation functors or ∞-categories, generalizing from characteristic 0 to positive characteristic and from deformations over local Artinian $k$-algebras to "derived" deformation functors, defined on DG analogs of Artinian $k$-algebras.

I have the following naive question about these generalizations.

Given a classical deformation problem in characteristic 0 (such as deformations of associative algebras or complex manifolds) "controlled" by a DG Lie or L_{∞} algebra $L$, in the sense that the deformation functor attached to this deformation problem (defined on the category of local commutative Artinian $k$-algebras) is isomorphic to the deformation functor attached to $L$, to what extent does $L$ encode the "derived" deformation theory of the original deformation problem?

What would be the steps and obstructions to turning the classical deformation functor associated to $L$ (from local commutative Artinian $k$-algebras to sets, say) into a "suped-up" version — for example as functor from the ∞-category of Artinian $\mathbb E_\infty$-algebras to the ∞-category of spaces as in Lurie's formulation of derived deformation theory?