The precession and damping of a collinear magnetization displaced from its equilibrium are well described by the Landau-Lifshitz-Gilbert equation. The theoretical and experimental complexity of noncollinear magnetizations is such that it is not known how the damping is modified by the noncollinearity. We use first-principles scattering theory to investigate transverse domain walls (DWs) of the important ferromagnetic alloy Ni80Fe20 and show that the damping depends not only on the magnetization texture but also on the specific dynamic modes of Bloch and Néel DWs in ways that were not theoretically predicted. Even in the highly disordered Ni80Fe20 alloy, the damping is found to be remarkably nonlocal.