  Let $\mathrm{Lip}(X)$, $\mathrm{Lip}^b(X)$,
  $\mathrm{Lip}^{\mathrm{loc}}(X)$ and $\mathrm{Lip}^\mathrm{pt}(X)$ be the
vector spaces of Lipschitz, bounded Lipschitz,
  locally Lipschitz and pointwise Lipschitz
  (real-valued) functions defined on a metric space $(X, d_X)$, respectively.
We show that if a weighted composition operator $Tf=h\cdot f\circ \varphi$
defines a bijection between such vector spaces preserving Lipschitz constants,
local Lipschitz constants or pointwise Lipschitz constants, then $h=
\pm1/\alpha$ is a constant function for some scalar $\alpha>0$ and $\varphi$ is
an $\alpha$-dilation.
  Let $U$ be open connected and $V$ be open, or both $U,V$ are convex bodies,
in normed linear spaces $E, F$, respectively. Let $Tf=h\cdot f\circ\varphi$ be
a bijective weighed composition operator between the vector spaces
$\mathrm{Lip}(U)$ and $\mathrm{Lip}(V)$, $\mathrm{Lip}^b(U)$ and
$\mathrm{Lip}^b(V)$, $\mathrm{Lip}^\mathrm{loc}(U)$ and
$\mathrm{Lip}^\mathrm{loc}(V)$, or $\mathrm{Lip}^\mathrm{pt}(U)$ and
$\mathrm{Lip}^\mathrm{pt}(V)$, preserving the Lipschitz, locally Lipschitz, or
pointwise Lipschitz constants, respectively. We show that there is a linear
isometry $A: F\to E$, an $\alpha>0$ and a vector $b\in E$ such that
$\varphi(x)=\alpha Ax + b$, and $h$ is a constant function assuming value $\pm
1/\alpha$. More concrete results are obtained for the special cases when
$E=F=\mathbb{R}^n$, or when $U,V$ are $n$-dimensional flat manifolds.
