A famous theorem of Gromov says that the set of compact Riemannian manifolds with $Ric \geq c$ and $\text{diam} \leq D$ is relatively compact in the Gromov-Hausdorff metric. Chapter 10 of the book by Burago-Burago-Ivanov lists a few other such compactness theorems, but they are all dependent on some variant of global curvature bounds.
My question is, are there examples of such compactness theorems which do not use assumptions on the curvature? In other words, is it reasonable to expect some kind of compactness theorem under uniform control of a combination of other geometric quantities like volumes of balls, curve lengths, diameter, injectivity radius etc., but not curvature?
This is mainly a reference request, I am trying to get a feel for whether there is a metamathematical principle which tells us that one cannot expect relative compactness in the Gromov-Hausdorff metric without curvature restrictions.
