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We consider the K\"ahler-Ricci flow on certain Calabi-Yau fibration, which isa Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yaufibrations with one dimensional bases. Assume the K\"ahler-Ricci flow on totalspace admits a uniform lower bound for Ricci curvature, then the flow convergesin Gromov-Hausdorff topology to the metric completion of the regular part ofgeneralized K\"ahler-Einstein current on the base, which is a compact lengthmetric space homeomorphic to the base. The analogue results for the continuitymethod on such Calabi-Yau fibrations are also obtained. Moreover, we show thecontinuity method starting from a suitable K\"ahler metric on the total spaceof a Fano fibration with one dimensional base converges in Gromov-Hausdorfftopology to a compact metric on the base. During the proof, we show the metriccompletion of the regular part of a generalized K\"ahler-Einstein current on aRiemann surface is compact.

Collapsing limits of the K\"ahler-Ricci flow and the continuity method