Assuming Flight within the northern hemisphere:
1. Magnetic needles suspended on pivot like a pendulum.
2. Lines of earth’s magnetic flux have horizontal and vertical component.
3. Horizontal component causes needles to align with local magnetic north.
4. Vertical component causes needles to dip slightly towards north.
5. Angle of dip causes the centre of gravity of the needle assembly to be displaced towards the south.
6. Any acceleration on an east or west heading will cause the c of g to swing rearwards, resulting in an apparent turn towards north.
7. Any deceleration on an east or west heading will cause the c of g to swing forwards, resulting in an apparent turn towards south.
8. Any acceleration or deceleration on a north or south heading will cause the c of g to nod forwards or backwards with no heading error.
9. During a steady turn, the effect of the acceleration on the displaced c of g is to cause an apparent underturn towards north and an apparent overturn towards south. There is no error on turns towards east or west.
As long as you understand that the c of g of the needle assembly is displaced by the vertical component of the earth’s magnetic field, and that the forces of turning and acceleration act directly on the pivot, the resulting effect on the c of g and therefore on the needle assembly should become clear.
Of course, a few diagrams would help but that’s beyond my computing ability!