Given two metric spaces we can define the Gromov-Hausdorff (GH) distance. There are compactness results stating that a sequence of manifolds of a fixed dimension, with a uniform lower Ricci bound and a uniform diameter upper bound (not necessary if we work we pointed-GH) GH-converges, up to a subsequence, to a metric space (see Petersen Theorem pag 300).

My question is what happens if we do not assume the uniform dimensional bound. In this case it is harder to recover Gromov's covering criterion (Petersen pag 299), but are there reasonable assumptions to have precompactness? Or, is there another notion of convergence that allows us to pass to infinite dimension?

For example, what can we say about the sequence of unit spheres $\mathcal{S}^n$?