Unimportant Question

OK, I know this doesn't make a bit of difference, but I was always that
kid that kept asking why and occasionally a strange question pops into
my head and I thought I'd throw it out for the flying gurus to take a
stab at if I can explain my question.

The AI works on the principle that a spinning gyro will remain in the
plane in which it spins, right? If taking a long flight, ie
transatlantic, why does flying around the curvature of the Earth not
affect this. The plane that parallels the ground in NYC won't be the
same plane that parallels the ground in London. Does gravity make it
precess gradually or what?