Sieve Theory seems like an example.
Originally developed for question related to twin-primes and Golbach-like questions, and other clearly number theoretic questions, such techniques are meanwhile used to address
other types of problems as well.
To illustrate this, as one example, let me mention the following result (Corollary 9.7) in a preprint of Kowalski, The principle of the large sieve:
Let $G$ be the mapping class group of a closed orientable surface of genus $g > 1$, let $S$ be a finite symmetric generating set of $G$ and let $(X_k)$, $k > 1$, be the simple left-invariant random walk on $G$. Then the set $X \subset G$ of non-pseudo-Anosov elements is transient for this random walk.
To get a more detailed impression, this preprint or (even better) the book by the same author, The Large Sieve and its Applications (CUP, 2008), into which this preprint developed, would be a good source.
