  We study a Fourier-Mukai kernel associated to a GIT wall-crossing for
arbitrarily singular (not necessarily reduced or irreducible) affine varieties
over any field. This kernel is closely related to a derived fiber product
diagram for the wall-crossing and simple to understand from the viewpoint of
commutative differential graded algebras. However, from the perspective of
algebraic varieties, the kernel can be quite complicated, corresponding to a
complex with multiple homology sheaves. Under mild assumptions in the
Calabi-Yau case, we prove that this kernel provides an equivalence between the
category of perfect complexes on the two GIT quotients. More generally, we
obtain semi-orthogonal decompositions which show that these categories differ
by a certain number of copies of the derived category of the derived fixed
locus. The derived equivalence for the Mukai flop is recovered as a very
special case.
