Operator Lipschitz functions in several variables and Möbius transformations

It is proved that if $f$ is an operator Lipschitz function defined on $\mathbb R^n$, then the function $\dfrac{f\circ\varphi}{\|\varphi'\|}$ is also operator Lipschitz for every Möbius transformations $\varphi$ with $f(\varphi(\infty))=0$. Here $\|\varphi'\|$ denotes the operator norm of the Jacobian matrix $\varphi'$.
Similar statements are obtained also for operator Lipschitz functions defined on closed subsets of $\mathbb R^n$.