It is well-known that there are Calabi-Yau's who are not birational but are derived equivalent. However I am interested in seeing D-equivalence as a weakening of birationality.
Q. If $X$ and $Y$ are smooth projective varieties and $X$ is birational to $Y$, does it follow that $D(X)\simeq D(Y)$?
My guess would be that no such result exists so far, but I cannot see a good obstruction to it happening. I know a lot of work is done to show that flips, flops and blowups are derived equivalent so it would mean that if the result above were true, the later would be easier to prove.