Hi, thanks for the tip. There is a Pre-AF setting which was turned on. This clearly didn't have much impact on my other (non-Sony) lenses, so I never noticed it before. Have turned it off, and the camera is no longer trying to focus on random things 🙂
Also, thanks for explaining what focus breathing is.
BTW, FE 20mm F1.8G is a cracking lens, really impressive optics.