I read Kawamata's note, and the goal of this note is to construct an equivalence $$ \Phi:D^{b}(X)\to D^{b}(Y)$$ for a flop $X\to Z\leftarrow Y$, and I wonder to the moduli approach and it seems to me that the claim of moduli approach is, $Y$ is the moduli of certain objects in $D^b(X)$.
My question is, if $(X,\Delta)$ is a klt pair, and $f:X\to Z$ is a $(K_X+\Delta)$-negative contraction which is small, is there any approach to construct the flip of $f$ using moduli approach? Of course, if $X$ is singular, $D^b(X)$ might not be a good object for $X$ as explained in the MO post, and we should use another object. Nevertheless, such approach seems good to me, especially if we see that the proof of (4.6) in this paper.
