ag.algebraic geometry - Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19

Fulton's "Intersection theory" book contains the following fact (example 18.3.19):

Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a dualizing sheaf $omega_Z$) and is of dimension $n$. If $E$ is locally free coherent sheaf on $X$, then:
$$tau_k(E) = (-1)^{n-k}tau_k(E^{vee}otimes omega_X) (*)$$
in $A_k(X)_{mathbb Q}$, the $k$-th Chow group of $X$ with rational coefficients. Here $tau: K_0(X) to A_*(X)_{mathbb Q}$ is the generalized Riemann-Roch homomorphism.

The formula follows from a more general one for complexes with coherent cohomology (and without Cohen-Macaulayness):
$$ sum (-1)^itau_k(mathcal H^i(C^{cdot})) = (-1)^ksum(-1)^itau_k(mathcal H^i(RHom(C^{cdot},omega^{cdot}_X))) (**)$$

In a proof I would like to use (*) in a more general setting:

Does anyone know a reference for (**) or (*) when $X$ is imbeddable in a regular scheme, not necessarily over a field (I am willing to assume $X$ is finite over some complete regular local ring)?

The original source of (**) (Fulton-MacPherson "Categorical framework for study of singular spaces") hints that a generalization is possible, then refers to Delign's appendix of Hartshorne "Residues and Duality"! We all know that fleshing out the details there is non-trivial, however.