Homological congruence formulae for characteristic classes of singular varieties

For a pair $(f, g)$ of morphisms $f:X \to Z$ and $g:Y \to Z$ of (possibly singular) complex algebraic varieties $X,Y,Z$, we present congruence formulae for the difference $f_*T_{y*}(X) -g_*T_{y*}(Y)$ of pushforwards of the corresponding motivic Hirzebruch classes $T_{y*}$. If we consider the special pair of a fiber bundle $F \hookrightarrow E \to B$ and the projection $pr_2:F \times B \to B$ as such a pair $(f,g)$, then we get a congruence formula for the difference $f_*T_{y*}(E) -\chi_y(F)T_{y*}(B)$, which at degree level yields a congruence formula for $\chi_y(E) -\chi_y(F)\chi_y(B)$, expressed in terms of the Euler--Poincarv'e characteristic, Todd genus and signature in the case when $F, E, B$ are non-singular and compact. We also extend the finer congruence identities of Rovi--Yokura to the singular complex projective situation, by using the corresponding intersection (co)homology invariants.