  We develop tools for proving isomorphisms of normed spaces of Lipschitz
functions over various doubling metric spaces and Banach spaces. In particular,
we show that
$\operatorname{Lip}_0(\mathbb{Z}^d)\simeq\operatorname{Lip}_0(\mathbb{R}^d)$,
for all $d\in\mathbb{N}$. More generally, we e.g. show that
$\operatorname{Lip}_0(\Gamma)\simeq \operatorname{Lip}_0(G)$, where $\Gamma$ is
from a large class of finitely generated nilpotent groups and $G$ is its
Mal'cev closure; or that
$\operatorname{Lip}_0(\ell_p)\simeq\operatorname{Lip}_0(L_p)$, for all $1\leq
p<\infty$.
  We leave a large area for further possible research.
