  We consider convex series of molecules in Lipschitz-free spaces, i.e.
elements of the form $\mu=\sum_n \lambda_n
\frac{\delta_{x_n}-\delta_{y_n}}{d(x_n,y_n)}$ such that $\|\mu\|=\sum_n
|\lambda_n |$. We characterise these elements in terms of geometric conditions
on the points $x_n$, $y_n$ of the underlying metric space, and determine when
they are points of G\^ateaux differentiability of the norm. In particular, we
show that G\^ateaux and Fr\'echet differentiability are equivalent for finitely
supported elements of Lipschitz-free spaces over uniformly discrete and bounded
metric spaces, and that their tensor products with G\^ateaux (resp. Fr\'echet)
differentiable elements of a Banach space are G\^ateaux (resp. Fr\'echet)
differentiable in the corresponding projective tensor product.
