There is a well-known theorem that between any two absolutely continuous Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}^n$ there is an increasing triangular
transformation $T : \mathbb{R}^n \to \mathbb{R}^n$, such that $ν = T_∗μ$ (see, for example, “Triangular
transformations of measures” by Bogachev, Kolesnikov, and Medvedev). 

There is another version of this theorem for smooth manifolds (non-compact Moser theorem).
If  $M$  is a  noncompact connected oriented manifold and  if $\mu$ and  $\nu$ are  volume forms on  M  with $\int_M \mu = \int_M \nu \le \infty$   and  if  each end  of  the manifold $M$  has finite $\mu$  volume if it has finite $\nu$  volume and  infinite $\mu$ volume if it  has  infinite $\nu$  volume, then there is  a  diffeomorphism $\phi: M \to M$,  such that  $\phi^* \mu = \nu$. (see "Diffeomorphisms and volume-preserving embeddings for noncompact manifolds" by Greene and Shiohama).

I am curious about the equivariant analog of either of these results. 

More precisely, assume that a (compact?) group $G$ acts on $\mathbb{R}^n$ and $\mu$ and $\nu$ are two $G$-invariant (smooth) densities. Does there always exist a $G$-equivariant diffeomorphism $T : \mathbb{R}^n \to \mathbb{R}^n$, such that $ν = T_∗μ$?

Could you point out if this result is proved somewhere in the literature? Does the group $G$ has to be compact for it to hold?

Thank you!