In K-stability, the characterization of K-stable varieties is well-studied when K_X is ample or X is a Calabi-Yau or Fano variety. However, K-stability of Fano fibrations or Calabi-Yau fibrations (i.e., K_X is relatively trivial) is not known much in algebraic geometry. On the other hand, cscK problems on fibrations are studied by Fine, Jian-Shi-Song and Dervan-Sektnan in Kahler geometry. We introduce adiabatic K-stability (If f:(X,H) to (B,L) is a fibration of polarized varieties, this means that K-stability of (X,aH+L) for sufficiently small a) and show that adiabatic K-semistability of Calabi-Yau fibration implies log-twisted K-semistability of the base variety by applying the canonical bundle formula. If the base is a curve, we also obtain a partial converse. In this talk, I would like to explain our main results and their applications to construction of their moduli and quasi-projectivity (ongoing work with Kenta Hashizume).