I am just a beginner in $D$-module, so this could be a stupid question, but I can't find an easy reference for it. I would like to define the notion of $D$-affine morphism. The most obvious way would be to say that a smooth morphism between smooth quasi-projective schemes:
$$ f : X \rightarrow Y$$
is *$D$-affine* if the following two conditions hold:

1) the functor $f_* : Mod_{qc}(D_X) \rightarrow Mod(f_*D_X)$ is exact,

2) for any $M \in Mod_{qc}(D_X)$, we have $f_*M = 0 \Rightarrow M=0$.

Since I want things to be somehow a bit functorial, I would like that the composition of two $D$-affine morphisms is still $D$-affine. But it's not clear to me that the direct image of a quasi-coherent $D$-module by an $D$-affine morphism is again a quasi-coherent $D$-module.

One could also define $D$-affiness by the following conditions:

1) the functor $\int_{X/Y}^{0} : Mod_{qc}(D_X) \rightarrow Mod_{qc}(D_Y)$ is exact,

2) for any $M \in Mod_{qc}(D_X)$, we have $\int _{X/Y}^0 M = 0 \Rightarrow M=0$.

here $\int_{X/Y}^0$ denotes the $0$-th homology of the derived push-forward for $D$-modules. Since $f$ is smooth the $\int_{X/Y}^k$ vanish for $k<0$, so that the Leray spectral sequence for $D$-modules push-forward guarantees me that the composition of two smooth $D$-affine morphisms is again $D$-affine. But with this definition, my favourites examples (that is $f : G/P \rightarrow spec \mathbb{C}$, for $G/P$ a rational homogeneous space) are not $D$-affine anymore.

Is there something I am missing?

Thanks a lot!