And I would imagine that gravity also accounts for the gyroscope artificial horizon phenomenon that flat-earthers also like to bring up. You would most likely know much more about it than myself, but it seems that the pull of gravity would keep the gyroscope perpendicular to the surface of the earth while an airplane is in flight, (I only bring it up because I saw it brought up in several other threads already and I do think that the answer to that particular objection of the flat-earthers pertains to the effects of gravity on the gyroscope or artificial horizon).
You're right. "Down" is a local phenomenon; it's just the direction of the center of the mass of the Earth. I haven't given it any thought before, but I suspect that a really sensitive gyroscope detects the gradual change in force. I wonder if those compasses are programmed to compensate for it.
The most devastating hit on a "flat Earth" belief is the Foucault pendulum. I've yet to see any flat-Earther come up with an even remotely plausible story to account for it.