Assume $G$ is a Lie group of dimension $n$ and $M\subset\mathbb{R}^m$. Consider the group action $G\times\mathbb{R}^m\to\mathbb{R}^m$. Consider now the $p$-fold product $M^p\subset\mathbb{R}^{mp}$ and the map $\mu:G\times\mathbb{R}^{mp}\to\mathbb{R}^{mp}$ defined as the diagonal action: $\mu(g,x_1,\dots,x_p)=(gx_1,\dots,gx_p)$.
What can we say about the dimension of $\mu(G\times M)$? Can we bound it?
This question is a follow-up of An example of a similar universal cover for 5 points.