Let $A$ be a Grothendieck abelian category. I will say that $A$ is of global dimension less or equal to $n$ if $Ext^{k}_{A}(a, b) = 0$ for $k > n$ and all $a, b \in A$. This is equivalent to saying that any object of $A$ admits an injective resolution of length at most $n$.

Let $I$ be a small diagram category, so that the category of functors $Fun(I, A)$ is again Grothendieck. Are there any general bounds on the dimension of this functor category in terms of the dimension of $A$ and some invariant of the diagram category?

Note that I do not expect this dimension would be finite for arbitrary $I$, I am rather looking for examples of "nice" $I$ for which we have such a bound for arbitrary Grothendieck category. In the simplest possible case, I would like to know the answer even if $A$ is a category of modules over a ring.