The following, if true, should probably be "standard," but I don't know where to look. I'd rather be slightly imprecise about hypotheses in the hope that there's a good general answer. Feel free to tell me that somewhat stronger hypotheses are needed.
Let $D$ be a pretriangulated dg category equipped with a $t$-structure for which the heart is an abelian category $A$. Let $T\subset A$ be a localizing subcategory, so in particular the quotient abelian category $A/T$ exists. Let $C\subset D$ be the full subcategory of $D$ consisting of objects whose cohomologies (with respect to the given $t$-structure) lie in $T$.
Belief: $A/T$ is equivalent to the heart of $D/C$ (w.r.t. induced $t$-structure).
Under what hypotheses is it true? What's a reference?