  Let $B$ be a smooth projective curve and let $\pi: \mathcal{X} \to B$ be a
smooth integral model of a geometrically integral Fano variety over $K(B)$.
Geometric Manin's Conjecture predicts the structure of the irreducible
components $M \subset \textrm{Sec}(\mathcal{X}/B)$ which parametrize
non-relatively free sections of sufficiently large anticanonical degree. Over
the complex numbers, we prove that for any such component $M$ the sections come
from morphisms $f: \mathcal{Y} \to \mathcal{X}$ such that the generic fiber of
$\mathcal{Y}$ has Fujita invariant $\geq 1$. Furthermore, we prove that there
is a bounded family of morphisms $f$ which together account for all such
components $M$. These results verify the first part of Batyrev's heuristics for
Geometric Manin's Conjecture over $\mathbb{C}$. Our result has ramifications
for Manin's Conjecture over global function fields: if we start with a Fano
fibration over a number field and reduce mod $p$, we obtain upper bounds of the
desired form by first letting the prime go to infinity, then the height.
