  Pruning processes $(\mathcal{F}(\theta),\theta\geq 0)$ have been studied
separately for Galton-Watson trees and for L\'evy trees/forests. We establish
here a limit theory that strongly connects the two studies. This solves an open
problem by Abraham and Delmas, also formulated as a conjecture by L\"ohr,
Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson
forests $\mathcal{F}_n$, $n\geq 1$, in the domain of attraction of a L\'evy
forest $\mathcal{F}$, suitably scaled pruning processes
$(\mathcal{F}_n(\theta),\theta\geq 0)$ converge in the Skorohod topology on
cadlag functions with values in the space of (isometry classes of) locally
compact real trees to limiting pruning processes. We separately treat pruning
at branch points and pruning at edges. We apply our results to study ascension
times and Kesten trees and forests.
