The kinetics of the flux of a charged macromolecular solution through an environment of changing geometry with wide and constricted regions is investigated analytically. A model device consisting of alternating deep and shallow slits known as an "entropic trap" is used to represent the environment. The flux is supported by the external electrostatic field. The "wormlike chain" model is used for the macromolecule (dsDNA in the present study). The chain entropy in both the deep and the shallow slits, the work by the electric field, and the energy of the elastic bending of the chain are taken into account accurately. Based on the calculated free energy, the kinetics and the scaling behavior of the chain escaping from the entropic trap are studied. We find that the escape process occurs in two kinetic stages with different time scales and discuss the possible influence of the surface roughness. The scope of the accuracy of the proposed model is discussed.