Searches for gravitational waves from compact binaries focus mostly on quasicircular motion with the rationale that wave emission circularizes the orbit. Here, we study the generality of this result, when astrophysical environments (e.g., accretion disks) or other fundamental interactions are taken into account. We are motivated by possible electromagnetic counterparts to binary black hole coalescences and orbits, but also by the possible use of eccentricity as a smoking gun for new physics. We find that (i) backreaction from radiative mechanisms, including scalars, vectors and gravitational waves, circularize the orbital motion. (ii) By contrast, environmental effects such as accretion and dynamical friction increase the eccentricity of binaries. Thus, it is the competition between radiative mechanisms and environmental effects that dictates the eccentricity evolution. We study this competition within an adiabatic approach, including gravitational radiation and dynamical friction forces. We show that there is a critical semimajor axis below which gravitational radiation dominates the motion and the eccentricity of the system decreases. However, the eccentricity inherited from the environment-dominated stage can be substantial, and in particular can affect LISA sources. We provide examples for GW190521-like sources.