In differential geometry, the use of geometric flows seem to fit in this framework. One could see Hamilton-Perelman proof as an instance of this phenomenon, but maybe the lack of canonicity of Ricci Flow with surgery disqualifies it.
However, Hamilton's theorem on 3-manifolds with $Ric>0$ and Brendle-Schoen differentiable sphere theorem would be interesting examples maybe. In these case, Ricci Flow (without surgery) creates a deformation between your initial metric (with $Ric>0$ or strictly quarter pinched) and metric of constant curvature $1$, since only quotients of the sphere bear such metrics, the initial manifold was a quotient of a sphere.
I'm sure there are also interesting examples in harmonic map heat flow (starting with the work of Eels and Sampson) but I don't know them well enough.
