We consider a ferromagnetic nanowire and we focus on an asymptotic regime where the Dzyaloshinskii-Moriya interaction is taken into account. First we prove a dimension reduction result via Γ-convergence that determines a limit functional E defined for maps m:R→S2 in the direction e1 of the nanowire. The energy functional E is invariant under translations in e1 and rotations about the axis e1. We fully classify the critical points of finite energy E when a transition between −e1 and e1 is imposed; these transition layers are called (static) domain walls. The evolution of a domain wall by the Landau-Lifshitz-Gilbert equation associated to E under the effect of an applied magnetic field h(t)e1 depending on the time variable t gives rise to the so-called precessing domain wall. Our main result proves the asymptotic stability of precessing domain walls for small h in L∞([0,+∞)) and small H1(R) perturbations of the static domain wall, up to a gauge which is intrinsic to invariances of the functional E.