Let $\mathbb{D}_n^d$ be the $d$-dimensional unit ball with $n$ punctures. I am interested in the groups of orientation-preserving diffeomorphisms $Diff(\mathbb{D}_n^d)$ that fix the punctures. In particular I want to know what we know about all fundamental groups of this space $$\pi_k(Diff(\mathbb{D}_n^d))$$
E.g. I would assume it is zero for $k\geq d$? Also for $k<d-1$? Or: Is there a good presentation via action on the cohomology $H^k(\mathbb{D}_n^d)$ or even the homotopy groups (which I know are not known).

The example $d=2$ is the braid group in $n$ strands, which acts on $\mathbb{Z}^n$ (cohomology) and moreover on the free group in $n$ generators. This is what I have in mind and want to understand for higher $d$.

Thanks for your help! And apologies, that my knowledge in algebraic topology is limited. Even more I appreachiate any hint you may have! Greetings, Simon