  We solve some open problems regarding diameter two properties within the
class of Banach spaces of real-valued Lipschitz functions by using the de Leeuw
transform. Namely, we show that: the diameter two property, the strong diameter
two property, and the symmetric strong diameter two property are all different
for these spaces of Lipschitz functions; the space $\operatorname{Lip}_0(K_n)$
has the symmetric strong diameter two property for every $n\in \mathbb{N}$,
including the case of $n=2$; every local norm-one Lipschitz function is a
Daugavet point.
