We present a new incremental mesh morphing method obtained by proposing and discretizing a solution procedure for the continuous morphing problem. Our method seeks a diffeomorphism that transforms an initial domain to a final domain by only prescribing the boundary displacement. To this end, we propose to minimize the distortion of the morphing mapping constrained to satisfy the imposed boundary displacement. To solve this problem, we consider an augmented Lagrangian method in Hilbert spaces that incorporates the boundary condition in the objective function using the Lagrange multipliers and a penalty parameter. The distortion is devised to penalize the appearance of non-invertible mappings and therefore, we do not need to equip our discrete implementation with untangling capabilities. Moreover, we introduce a weight function to improve the quality of the deformation and thus, the robustness of the non-linear solver. The discretization of the continuous augmented Lagrangian method leads to a mesh morphing method suitable for large displacements and rotations of meshes with non-uniform sizing, and mesh curving of highly stretched high-order meshes.