Odaka proposed the K-moduli conjecture in 2010, predicting the existence of a moduli space of K-polystable objects with an ample CM line bundle. While this conjecture has been solved in the Fano case, it remains open in general. Recent developments of Fine, Dervan-Sektnan and Ortu have highlighted the relevance of the existence of cscK metrics and K-stability for $(X,\epsilon A+L)$ for sufficiently small $\epsilon$, where $f\colon (X,A)\to (B,L)$ is a fibration. According to their works, such K-stability is closely related to some K-stability of fibers and the bases. Especially in the Calabi-Yau fibration over curve case, uniform K-stability in this context (uniform adiabatic K-stability) coincides with the log twisted K-stability on the base. In this talk, we will regard the base curve as a quasimap and introduce the notion of K-moduli of quasimaps. By using this framework, we address the K-moduli conjecture for Calabi-Yau fibrations over curves whose generic fibers are either Abelian varieties or HyperKahler manifolds. This is a joint work arXiv:2504.21519 with Kenta Hashizume.