Branner–Hubbard motion is a systematic way of deforming an attracting holomorphic dynamical system f into a family (fs)s ∈ L, via a holomorphic motion which is also a group action. We establish the analytic dependence of fs on s (a result first stated by Lyubich) and the injectivity of fs on f. We prove that the stabilizer of f (in terms of s) is either the full group L (rigidity), or a discrete subgroup (injectivity). The first case means that fs is Möbius conjugate to f for all s ∈ L, and it happens for instance at the center of a hyperbolic component. In the second case the map s ↦ fs is locally injective. We show that BH-motion induces a periodic holomorphic motion on the parameter space of cubic polynomials, and that the corresponding quotient motion has a natural extension to its isolated singularity. We give another application in the setting of Lavaurs enriched dynamical systems within a parabolic basin.