Bivariant theories of constructible functions and Grothendieck transformations

We discuss several reasonable bivariant theories of constructible functions and discuss the existence or non-existence of Grothendieck transformations to bivariant homology theories, motivated by the operational bivariant theory introduced by Fulton and MacPherson. By analyzing these bivariant theories, we give a theorem concerning a certain Grothendieck transformation to the operational bivariant theory.