The paper [Chromatic homotopy theory is asymptotically algebraic][1] (by Barthel, Schlank and Stapleton) is an interesting example.  In chromatic homotopy theory we study the category $\mathcal{L}(p,n)$ of spectral local with respect to the Johnson-Wilson theory $E(p,n)$, where $p$ is a prime and $n$ is a positive integer.  It is a folklore idea that when $p$ is large relative to $n$, the properties of $\mathcal{L}(p,n)$ become more and more algebraic and more and more independent of $p$, in some sense.  The cited paper formalises this idea by taking a kind of ultraproduct over primes.  There is also a sequel called [Monochromatic homotopy theory is asymptotically algebraic][2].


  [1]: https://arxiv.org/abs/1711.00844
  [2]: https://arxiv.org/abs/1903.10003