  We use augmented commutative differential graded algebra (ACDGA) models to
study $G$-representation varieties of fundamental groups $\pi=\pi_1(M)$ and
their embedded cohomology jump loci, around the trivial representation 1. When
the space $M$ admits a finite family of maps, uniformly modeled by ACDGA
morphisms, and certain finiteness and connectivity assumptions are satisfied,
the germs at 1 of ${\rm Hom} (\pi,G)$ and of the embedded jump loci can be
described in terms of their infinitesimal counterparts, naturally with respect
to the given families. This approach leads to fairly explicit answers when $M$
is either a compact K\"ahler manifold, the complement of a central complex
hyperplane arrangement, or the total space of a principal bundle with formal
base space, provided the Lie algebra of the linear algebraic group $G$ is a
non-abelian subalgebra of $\mathfrak{sl}_2(\mathbb{C})$.
