  Let $E$ be an elliptic surface over the curve $C$, defined over a number
field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For
any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of
(algebraic) degree at most $D$ over $k$, such that $\ell^n Q=P_t$, for some
$n\geq 1$. The bound obtained depends only on $\ell$, the surface and section
in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all
fibres of bounded degree. In special cases, we obtain more specific, in some
instances sharp, bounds.
