Given a manifold M and a point in its interior, the point-pushing map describes a diffeomorphism 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 describe for any d≥3 and k≥1 the action, by homotopy classes of homotopy equivalences, of the k-th braid group of M on the k-punctured manifold M ∖ z. 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 on the homology of configuration-mapping spaces, we describe the action of the braid group of M on the fibres of configuration-mapping spaces.