  Let $X$ be a del Pezzo surface over the function field of a complex curve. We
study the behavior of rational points on $X$ leading to bounds on the counting
function in Geometric Manin's Conjecture. A key tool is the Movable Bend and
Break Lemma which yields an inductive approach to classifying relatively free
sections for a del Pezzo fibration over a curve. Using this lemma we prove
Geometric Manin's Conjecture for certain split del Pezzo surfaces of degree
$\geq 2$ admitting a birational morphism to $\mathbb P^2$ over the ground
field.
