Curved mesh generation starting from a P1 mesh and closed advancing-layer boundary layer mesh generation both rely on mesh deformation and mesh optimization techniques. The approach presented in this work is to generalize connectivity-change moving mesh methods to high-order meshes. This approach is based on a high-order linear elasticity solver for the mesh deformation and on high-order mesh optimization operators such as mesh smoothing and generalized swapping. Thanks to this method, Pk meshes are generated from P1 meshes and closed advancing-layer boundary layer mesh generation will soon be possible.