  We establish a general computational scheme designed for a systematic
computation of characteristic classes of singular complex algebraic varieties
that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly
geometric and of a recursive nature terminating on genera of explicit
characteristic subvarieties that we construct. It enables us e.g. to apply
intersection theory of Schubert varieties to obtain a uniqueness result for
such characteristic classes in the homology of an ambient Grassmannian. Our
framework applies in particular to the Goresky-MacPherson L-class by virtue of
the Gysin restriction formula obtained by the first author in previous work. We
illustrate our approach for a systematic computation of the L-class in terms of
normally nonsingular expansions in examples of singular Schubert varieties that
do not satisfy Poincar\'{e} duality over the rationals.
