Proposition: Let $f:X\rightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds, and $L$ be a holomorphic line bundle. Then there exists a Zariski open set $Y_0\subset Y$ such that for any $y\in Y_0$, we have $$f_*(K_{X/Y}+L)_y=H^0(X_y,K_{X_y}+L|_{X_y}), $$ where $X_y=f^{-1}\{y\}$.

I want to whether this statement is true, and if it is, how to prove this.

Another question: Since for any sheaf $\mathcal{A}$ defined on $X$, we know that $f_*(\mathcal{A})_y=H^0(X_y, \mathcal{A}|_{X_y})$ for any $y\in Y$. If we replace $\mathcal{A}$ by $K_{X/Y}+L$, why we could not obtain the formula in the proposition?

Thank you for your attention.