Non-quasistationary evolution of the nearly limiting Gardner-equation solitons, which is caused by the variable parameters of the medium is considered. The proposed approximate description of such a process is based on representing the Gardner-equation solitons as compound formations formed by different-polarity kinks. The equations describing evolution of both the field drops in the kinks and the slowly varying (compared with the drops) fields connecting the kinks are derived. The problem of the soliton evolution in the case of a linear (in time) change of the cubic nonlinearity coefficient of the Gardner equation is solved analytically. The obtained approximate solution is compared with the results of the previous direct numerical integration of this problem.