  In this article, we study the higher-order regularity of the K\"ahler-Ricci
flow on compact K\"ahler manifolds with semi-ample canonical line bundle. We
proved, using a parabolic analogue of Hein-Tosatti's work on collapsing
Calabi-Yau metrics, that when the generic fibers of the Iitaka fibration are
biholomorphic to each other, the flow converges locally smoothly away from
singular fibers to a negative K\"ahler-Einstein metric on the base manifold. In
particular, we proved that the Ricci curvature of the flow is uniformly bounded
on any compact subsets away from singular fibers when the generic fibers are
biholomorphic to each other.
