Stephen Keith used the Gromov-Hausdorff convergence to study the existence of (measurable) differentiable structure on metric measure spaces that supports a Poincare inequality or K-Lip-lip condition. 

Juha Heinonen, Jeff Cheeger and Stephen keith also used this method as a standard blow up argument in related questions. 

Heinonen, Juha; Keith, Stephen Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds. Publ. Math. Inst. Hautes Études Sci. No. 113 (2011), 1–37. 

Keith, Stephen A differentiable structure for metric measure spaces. Adv. Math. 183 (2004), no. 2, 271–315.

J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (3) (1999) 428–517.

J. Cheeger, T.H. Colding, On the structure of spaces with Ricci curvature bounded below. I, II, III, J. Differential Geom.