We consider the dynamics of one-dimensional micromagnetic domain walls in layers of uniaxial anisotropy. In the regime of bulk materials, i.e. when the thickness is assumed to be infinite, and the magnetostatic interaction terms appear as local quantities, explicit traveling wave solutions for the corresponding Landau-Lifshitz equation, known as Walker exact solutions, can be constructed. A natural question is whether this construction can be perturbed to the non-local regime of layers of finite thickness. Our stability analysis gives an affirmative answer.