You are partly correct. KVL is indeed an approximation to Maxwell's equations based on the assumptions of circuit theory which are, as follows:
1) the circuit is small relative to the wavelengths involved (lumped-parameter system)
2) no net charge on any component at any time
3) no magnetic coupling between components
You can have changing magnetic fields within a component (e.g. in an inductor) as long as there is no magnetic coupling between components.