Looking for generalization of projective model structure

We can apply the Kan transfer theorem (Theorem 11.3.2 in Hirschhorn) to the right adjoint functor U: M^C→M^D, where M^D is equipped with the projective model structure. Its left adjoint F sends representable functors y(X) on D to the corresponding representable functors y(X) on C. The conditions in the theorem boil down to verifying that U takes relative F(y(X)⊗J)-complexes to weak equivalences (J denotes a set of generating acyclic cofibrations of M), where X∈D.

However, F(y_D(X)⊗J)=y_C(X)⊗J is a subclass of projective acylic cofibrations (which are generated by y_C(X)⊗J for all X∈C), and every projective acyclic cofibration is an objectwise weak equivalence, which completes the proof.

This model structure has appeared explicitly in the paper by Paul Balmer and Michel Matthey "Codescent theory I: Foundations". Theorem 3.5 establishes the existence of, what the authors called, the relative model structure.

But the ideas behind this model structure go back to Dwyer-Kan concept of an orbit introduced in the paper "Singular functors and realization functors". If you choose the set of orbits to be the representable functors $hom_{\mathcal{C}}(d,-)$ for all $d\in \mathcal{D}$, then this set of orbits defines the same $\mathcal{D}$-relative model structure on the category of $\mathcal{C}$-indexed diagrams.

You might be thinking of the paper "On Modified Reedy and Modified Projective Model Structures" by Mark Johnson. It does exactly what you asked in your question, in Proposition 6.4. Funny you should mention reinventing the wheel; I reproved this result a few years ago in a mathoverflow answer, before I knew about Mark's paper.