In this paper, we investigate the behaviour of the Serre spectral sequence
with respect to the algebraic structures of string topology in generalized
homology theories, specificially with the Chas-Sullivan product and the
corresponding coproduct and the module structures. We prove compatibility for
two kinds of fibre bundles: the fibre bundle $\Omega^n M \to L^n M \to M$ for
an h_*-oriented manifold M and the looped fibre bundle $L^n F \to L^n E \to L^n
B$ of a fibre bunde $F \to E \to B$ of h_*-oriented manifolds. Our method lies
in the construction of Gysin morphisms of spectral sequences. We apply these
results to study the ordinary homology of the free loop spaces of sphere
bundles and generalized homologies of the free loop spaces of spheres and
projective spaces.