  We introduce convex integrals of molecules in Lipschitz-free spaces
$\mathcal{F}(M)$ as a continuous counterpart of convex series considered
elsewhere, based on the de Leeuw representation. Using optimal transport
theory, we show that these elements are determined by cyclical monotonicity of
their supports, and that under certain finiteness conditions they agree with
elements of $\mathcal{F}(M)$ that are induced by Radon measures on $M$, or that
can be decomposed into positive and negative parts. We also show that convex
integrals differ in general from convex series of molecules. Finally, we
present some standalone results regarding extensions of Lipschitz functions
which, combined with the above, yield applications to the extremal structure of
$\mathcal{F}(M)$. In particular, we show that all elements of $\mathcal{F}(M)$
are convex series of molecules when $M$ is uniformly discrete and identify all
extreme points of the unit ball of $\mathcal{F}(M)$ in that case.
