Given a manifold M and a point in its interior, the point-pushing map describes a dif- feomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of M to the group of isotopy classes of diffeomorphisms of M that fix the basepoint. This map is well-studied in dimension d = 2 and is part of the Birman exact sequence. Here we study, for any d 􏰄 3 and k 􏰄 1, the map from the k-th braid group of M to the group of homotopy classes of homotopy equivalences of the k-punctured manifold M 􏰇 z, and analyse its injectivity. Equivalently, we describe the monodromy of the universal bundle that associates to a configuration z of size k in M its complement, the space M 􏰇 z. Furthermore, motivated by our work in [PT21], we describe the action of the braid group of M on the fibres of configuration-mapping spaces.