  We develop the technique of weight truncation in the context of
wall-crossings in birational cobordisms, parallel to that in [HL15, BFK19].
More precisely, for each such wall-crossing, we embed the bounded above derived
category of coherent sheaves of the semistable part as a semi-orthogonal
summand of that of the stack in question. Our construction does not require any
smoothness assumptions, and exhibits a strong symmetry across the two sides of
the wall-crossing. As an application, we show that for wall-crossings
satisfying suitable regularity conditions, a certain duality of local
cohomology complexes implies the existence of a fully faithful
functor/equivalence between the derived categories under wall-crossings.
