One important way to study a variety is to study the curves on it. Free curves, those whose restricted tangent bundle has vanishing H^1, are particularly nice. They are smooth points of the space Hom(B,X) of morphisms from B to X, and they lie on a unique component of Hom(B,X) of the expected dimension. Nonfree curves, on the other hand, are more mysterious, and the components of Hom(B,X) consisting entirely of nonfree curves are particularly challenging to study. In this talk, we provide a geometric classification of the ways that a family of curves can be nonfree. In the more general setting of a Fano fibration, we show that these curves come from a-covers of X, and we show that the set of such a-covers up to equivalence form a bounded family. This proves the first of Batyrev's heuristics that make up Geometric Manin's Conjecture for Fano fibrations over arbitrary base curves and provides strong evidence for Lehmann, Sengupta and Tanimoto's proposed characterization of the thin set in Manin's Conjecture. This is joint work with Brian Lehmann and Sho Tanimoto.