There are some finite-dimensional variants of Craig's example.
In the world of diffeomorphism groups of manifolds, Smale's theorem is that the group of diffeomorphisms of the 2-disc which restrict to the identity on the boundary circle, this is a contractible space. Denote this group by $Diff(D^2)$. There are fibre-bundles:
$$ Diff(D^2 \text{ fix } P_n) \to Diff(D^2) \to C_n(D^2)$$
where $C_n(D^2)$ is the configuration space of $n$ distinct points in the interior of $D^2$, and $P_n \subset D^2$ is an $n$-point subspace of $D^2$. This is how one sees the configuration space of $n$ distinct points in the disc as the classifying space of the pure braid group, which is $\pi_0 Diff(D^2 \text{ fix } P_n)$. Similarly,
$$ Diff(D^2, P_n) \to Diff(D^2) \to C_n(D^2)/\Sigma_n$$
is a fibre bundle, giving the unordered configuration space as the classifying space of the full braid group.
There are similar results in dimension $3$, where instead of Smale's theorem, you use Hatcher's, that $Diff(D^3)$ is contractible. So you can do similar things for configuration spaces, but in this case $Diff(D^3, P_n)$ is not homotopy-discrete if $n >1$. You can also make similar constructions for knots. Given a smooth embedding $f : [-1,1] \to D^3$ which agrees with the standard inclusion $x \longmapsto (x,0,0)$ on the boundary, you get fibre-bundles
$$Diff(D^3 \text{ fix } f([-1,1])) \to Diff(D^3) \to \mathcal{K}_{3,1}(f)$$
here $\mathcal K_{3,1}$ is the space of all knots in $D^3$, and $\mathcal K_{3,1}(f)$ is the path-component of $f$ in $\mathcal K_{3,1}$.
In this case the fiber is a homotopy-discrete group, and the groups can be fairly elaborate. In general, they're all finitely covered by groups that are products of braid groups, but the covers can be arbitrarily large. These groups have a somewhat similar feel to subgroups of the braid group that all preserve a common reduction curve system (in the sense of Thurston's classification of surface automorphisms).