  We show that, based on the critical state model for flux-line pinning in hard
superconductors, one can assess the magnetic moment relaxation induced by the
oscillations of a perpendicular magnetic field. Our theory follows a recent
proposal of using phenomenological 2D modeling for the description of crossed
field dynamics in high-T$_c$ superconductors [{\tt arXiv:cond-mat/0703330}].
  Stationary regimes with either saturation to metastable configurations, or
complete decay to the thermodynamic equilibrium are obtained. The transition
between both types of response is related to the disappearance of a flux free
core within the sample. As a common feature, a step-like dependence in the time
relaxation is predicted for both cases. The theory may be applied to long bars
of arbitrary and non homogeneous cross section, under in-plane magnetic field
processes.
