I think, Milnor's exotic spheres are an example (see his 1956 article). He uses Morse theory to deduce the homeomorphism between the exotic spheres. But the gradient flow used in Morse theory is not really explicit, but only a solution of some differential equation.

Similar in spirit are many diffeomorphism/homeomorphism proofs in algebraic topology. A standard tool is the h-cobordism theorem (by Smale and Freedman) which tells you that if for two manifolds X and Y of dimension greater than 3, there is a cobordism W such that $X \to W$ and $Y\to W$ are homotopy equivalences and $X$, $Y$ are simply connected, then $X$ and $Y$ are homeomorphic. If the dimension is greater than 4, even diffeomorphic. This homeomorphism/diffeomorphism is not really explicit either.