Let $M = \mathbb{R}^2, N = D^2$. In both cases I think the configuration space of $k$ distinct unordered points has the homotopy type of $K(B_k, 1)$. It would be interesting to find an example where $M$ and $N$ are both closed manifolds.

Let $M = \mathbb{R}^2, N = D^2$. In both cases I think the configuration space of $k$ distinct unordered points has the homotopy type of $K(B_k, 1)$. More generally I think we can take $N$ to be a manifold with boundary and $M = N \setminus \partial N$, and there should be some reasonable generalization of this but I can't think of a precise statement. It would be interesting to find an example where $M$ and $N$ are both closed manifolds (or show that one doesn't exist!).