Worth resurrecting I thought as @JRW87's comment deserves another attempt at an answer.
In all cases the bullet in flight (other influences neglected) does maintain the same eastward component of motion as the rifle it was fired from, just like the peanut in the plane example. The shooter and target travel at the same angular velocity, but if they are at different latitudes - as they would be for shots taken to the north or south – they would not be moving at the same linear speed since the radii to the axis of rotation will differ. It is this difference that makes the target move ahead (shooting towards the equator) or lag behind (away from equator) the bullet's path.
For directly east/west shots, where the linear speed is the same, then the reason for there also being left/right movement comes from the imbalance to original centripetal acceleration where bullets shot eastwards drop less than expected, and those to the west more. This movement is at right angles to the axis of rotation, so is not actually normal to the earth’s surface except at the equator, so skewed. This can be split into two components acting at right-angles, one which does act in the same axis as gravity - the Eotvos or vertical deviation from the expected gravity-based trajectory - and a horizontal component. Bullets shot to the east drift towards the equator from this horizontal component. The reverse applies for shots taken west.
All shots away from the cardinal compass points are influenced by a mix of both effects, but they all deliver left drift in the southern hemisphere. The difference in the size of deflection with direction affects long range trajectories. I've read that treating all directions the same is acceptable for shooting out to a kilometre or two. An interesting effect even if the resulting variations in POI are small enough to disregard.




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