  We show that the Lipschitz-free space $\mathcal{F}(X)$ over a superreflexive
Banach space $X$ has the property that every weakly precompact subset of
$\mathcal{F}(X)$ is relatively super weakly compact, showing that this space
"behaves like $L_1$" in this context. As consequences we show that
$\mathcal{F}(X)$ enjoys the weak Banach-Saks property and that every subspace
of $\mathcal{F}(X)$ with nontrivial type is superreflexive. Further, weakly
compact subsets of $\mathcal{F}(X)$ are super weakly compact and hence have
many strong properties. To prove the result, we use a modification of the proof
of weak sequential completeness of $\mathcal{F}(X)$ by Kochanek and Perneck\'a
and an appropriate version of compact reduction in the spirit of Aliaga,
No\^us, Petitjean and Proch\'azka.
