What actually moves Earth's magnetic north pole?

Answer: Churning liquid iron in the outer core

Churning liquid iron in the outer core ✓ — Correct! Earth's magnetic field is generated by convection in the outer core — molten iron swirling about 2,900 km below the surface. When those flows shift, the magnetic pole shifts. The north pole is currently racing from Canada toward Siberia at ~35 km/year, driven by a tug-of-war between field-generating patches deep beneath each region.

Melting polar ice shifting Earth's mass — Not quite. Ice melt does redistribute mass and nudges Earth's rotation axis — the GEOGRAPHIC pole — by centimeters per year. But that's independent of the MAGNETIC pole, which lives thousands of kilometers down in the liquid iron core.

The Sun's magnetic field tugging on it — Not quite. The Sun's magnetic field, carried by the solar wind, compresses Earth's magnetosphere on the dayside and stretches it into a long tail on the nightside. Those interactions shape the field's outer edge, but they don't reach the deep core where the field is actually generated.