Question

Why does a charge movind inside a cavity not change the outer charge distribution?

Answer 1

The field of the induced charges is not used up to cancel the field in the metal of the inner charges, so that it could no longer be used for anything else.

The two effects (field of the charges inside the conductor and the external field) are independent. Each one accumulates charges on the outside of the conductor to kill the field inside of it. The effects are additive, so applying an additional external field simply will displace more charges to the surface of the conductor.

Mathematically speaking, you find the reason for why the effects are additive in the linearity of Maxwells equations. This is also the reason why the principle of superposition works

Answer 2

Let us take an arbitrary conductor having a weird-shaped cavity inside it. Let +q charge be inserted inside the cavity. The field of +q attracts negative charge & repels positive charge; negative charge accumulates on the surface of the cavity while the induced positive charge accumulates on the outer surface of the conductor till the electric field of the +q charge in the cavity is balanced by the field due to the induced charge inside the conductor. Hence, there exists no electric field inside the conductor due to cancellation of the electric field of the charge residing in the cavity & that formed by the induced charges.