Many people have applied Gromov-Hausdorff convergence to obtain information about Riemannian manifolds with nonnegative Ricci curvature. Controls on Gromov-Hausdorff limits can lead to controls on diameters. For example in my thesis I proved, among other things, that manifolds with nonnegative Ricci curvature and linear volume growth have sublinear diameter growth. The proof uses Gromov-Hausdorff convergence and methods of Cheeger-Colding to obtain that convergence.

Other times Riemannian geometers use Gromov-Hausdorff convergence to come up with an idea but later simplify the proof in a way which circumvents actually mentioning the Gromov-Hausdorff convergence. I have a paper about fundamental groups of manifolds with nonnegative Ricci curvature and my original proof of the main theorem involved taking a Gromov-Hausdorff limit (as described in the final section of the paper). Then I thought of a simplification which allows one to obtain the main theorem without appealing to Gromov's compactness theorem.