  In this paper, we introduce the associated geodesic-Einstein flow for a
relatively ample line bundle $L$ over the total space $\mathcal{X}$ of a
holomorphic fibration and obtain a few properties of that flow. In particular,
we prove that the pair $(\mathcal{X}, L)$ is nonlinear semistable if the
{associated} Donaldson type functional is bounded from below and the
geodesic-Einstein flow has long-time {existence property}. We also define the
associated $S$-classes and $C$-classes for $(\mathcal{X}, L)$ and obtain two
inequalities between them when $L$ admits a geodesic-Einstein metric. Finally,
in the appendix of this paper, we prove that a relatively ample line bundle is
geodesic-Einstein if and only if an associated infinite rank bundle is
Hermitian-Einstein.
