Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories with enough injectives. This spectral squence converges under mild conditions, such as $\mathcal F$ maps injectives to $\mathcal G$-acyclics.
Let us now consider the most general version of this setting (that I am aware of). Let $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ be additive functors between abelian categories. In Proposition 5.4 of Hartshorne's "Residues and Duality" a canonical natural transformation from $R(\mathcal G \circ \mathcal F)$ to $(R \mathcal G)(R \mathcal F)$ is constructed, which is a natural isomorphism under mild hypotheses. Hartshorne also remarks that one can recover the Grothendieck spectral sequence from this proposition by taking cohomology and using the spectral sequence of a double complex.
Here's my question - let us now work in the most general setting. Let $\mathcal A$, $\mathcal B$, $\mathcal C$ be abelian categories and $\mathcal F$, $\mathcal G$ be additive functors as above. Suppose that the right derived functors $R \mathcal F$ and $R \mathcal G$ exist on the level of unbounded derived categories with the usual properties. What conditions on $R \mathcal F$ and $R \mathcal G$ ensure the convergence of the resulting spectral sequence for composition of functors? Can one say something in terms of hypercohomology spectral sequences for $R \mathcal F$ and $R \mathcal G$? More specifically, can one say that the spectral sequence for composition of functors converges if the hypercohomology spectral sequences for $R \mathcal F$ and $R \mathcal G$ converge?
