  We investigate a conjecture due to Haefliger and Thurston in the context of
foliated manifold bundles. In this context, Haefliger-Thurston's conjecture
predicts that every $M$-bundle over a manifold $B$ where $\text{dim}(B)\leq
\text{dim}(M)$ is cobordant to a flat $M$-bundle. In particular, we study the
bordism class of flat $M$-bundles over low dimensional manifolds, comparing a
finite dimensional Lie group $G$ with $\text{Diff}_0(G)$ and localizing the
holonomy of flat M-bundles to be supported in a ball.
