  We consider generalized (mixed) Monge-Amp\`ere products of
quasiplurisubharmonic functions (with and without analytic singularities) as
they were introduced and studied in several articles written by subsets of M.
Andersson, E. Wulcan, Z. B{\l}ocki, R. L\"ark\"ang, H. Raufi, J. Ruppenthal,
and the author. We continue these studies and present estimates for the Lelong
numbers of pushforwards of such products by proper holomorphic submersions.
Furthermore, we apply these estimates to Chern and Segre currents of
pseudoeffective vector bundles. Among other corollaries, we obtain the
following generalization of a recent result by X. Wu. If the non-nef locus of a
pseudoeffective vector bundle $E$ on a K\"ahler manifold is contained in a
countable union of $k$-codimensional analytic sets, and if the $k$-power of the
first Chern class of $E$ is trivial, then $E$ is nef.
