Let $S$ and $M$ be two finite-dimensional smooth manifolds with $\dim S\le \dim M$. Then it is known (e.g.Kriegl-Michor's book) that the set $\mathrm{Emb}(S, M)$ of all smooth embeddings $S\to M$ is an infinite-dimensional manifold; moreover, it is the total space of a smooth principal fiber bundle with structure group $\mathrm{Diff}(S)$ which has a base manifold $B(S,M)$ consisting of all submanifolds of $M$ of type $S$: $\require{AMScd}$ \begin{CD} \mathrm{Diff}(S) @>>> \mathrm{Emb}(S,M) \\ @. @VV{\pi}V \\ @. B(S,M) \end{CD}

Now, let's consider the diffeomorphism group $\mathrm{Diff}(M)$ of **the ambient space** $M$ rather than $S$. Then it seems that it has a natural action on $\mathrm{Emb}(S,M)$ (and even on $B(S,M)$) by
$$
\Big( \phi, (i:S\to M) \Big) \mapsto ( \phi\circ i :S\to M )
$$
where $\phi\in \mathrm{Diff}(M)$.

Question:It would be interesting to study the properties of this $\mathrm{Diff}(M)$ action on $\mathrm{Emb}(S,M)$, like what are the orbits, the orbit space, etc. Moreover, it seems that this action preserves the fibers, so it should induce an action on the base $B(S,M)$. Is this true? Anyway, I fail to find an answer in literature. Does anyone know any reference?