The paper presents a phenomenological theory of alternmagnets, a new class of collinear magnetic order, which captures their unique magnetization dynamics and allows for modeling magnetic textures in this phase. The theory explains the characteristic lifted degeneracy of magnon spectra in prototypical d-wave alternmagnets like RuO$_2$ by the emergence of an effective sublattice-dependent anisotropic spin stiffness. This stiffness leads to finite gradients of magnetization in alternmagnetic domain walls, even for 180° domain walls, which generate a ponderomotive force in the presence of inhomogeneous magnetic fields. The motion of these domain walls is characterized by an anisotropic Walker breakdown, with higher speed limits than ferromagnets but lower than antiferromagnets. The theory is supported by spin lattice model simulations and provides insights into the unique properties of alternmagnetic materials, such as the splitting of magnon modes and the behavior of domain walls.The paper presents a phenomenological theory of alternmagnets, a new class of collinear magnetic order, which captures their unique magnetization dynamics and allows for modeling magnetic textures in this phase. The theory explains the characteristic lifted degeneracy of magnon spectra in prototypical d-wave alternmagnets like RuO$_2$ by the emergence of an effective sublattice-dependent anisotropic spin stiffness. This stiffness leads to finite gradients of magnetization in alternmagnetic domain walls, even for 180° domain walls, which generate a ponderomotive force in the presence of inhomogeneous magnetic fields. The motion of these domain walls is characterized by an anisotropic Walker breakdown, with higher speed limits than ferromagnets but lower than antiferromagnets. The theory is supported by spin lattice model simulations and provides insights into the unique properties of alternmagnetic materials, such as the splitting of magnon modes and the behavior of domain walls.