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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.