Effective field theories (EFTs) facilitate what might otherwise be completely untenable calculations by helping us focus on only the most relevant physics at hand. Applied to general relativity, these techniques have famously improved our handle on post-Newtonian theory over the last decade, and extensions of these ideas are now also being developed to study how binary systems evolve when they are not isolated in empty space, but subjected to external perturbations. After discussing some of the key steps underpinning its construction, I will illustrate the capabilities of this EFT approach by using it to make novel predictions about how binary systems of Kerr (or Schwarzschild) black holes interact with ambient, light scalar fields. Despite being subject to the usual no-hair theorems, we will find that these systems can still exhibit interesting phenomenology due to the interplay between absorption at the horizons and momentum transfer in the bulk of the spacetime. These include the appearance of dynamically induced multipole moments, an emergent “fifth” force, and a superradiance-like mechanism fuelled by the binary’s orbital motion.