How to add performance benchmarks (like automated tests) to a project

I want to add performance benchmarks to the WordPress REST API, and I can’t find what are the standard tools for adding it to a software project, much less PHP.

I would have thought that just like you add PHPunit tests to a project, you could add some sort of other tests to verify the code is performant (ie that any new changes don’t cause it to run much slower.)

Any names of useful tools you could suggest? Or projects (especially PHP) that have benchmarking as part of their build process?

(I’ve used CacheGrind to check the performance of a single request, but now I’m wanting to benchmark the speed of lots of different functions, and make it easy for other contributors to do it, and possibly run them via Travis CI or something.)