  We use monopole Floer homology to study the topology of the space of contact
structures on a $3$-manifold which are isotopic to a fixed one. Our main tool
is a generalization of the contact invariant to an invariant of families, and
we establish a foundational result that describes its interaction with the
module structure in the Floer theory. We apply this to study the existence of
families of non-zero degree for several examples, such as Brieskorn homology
spheres and surgeries on the Figure Eight knot. More generally, our techniques
provide obstructions to the homological triviality of a natural fibration over
the $2$-sphere with total space given by the space of contact structures.
