  We investigate resonance crossings of a charged body moving around a Kerr
black hole immersed in an external homogeneous magnetic field. This system can
serve as an electromagnetic analogue of a weakly non-integrable extreme mass
ratio inspiral (EMRI). In particular, the presence of the magnetic field
renders the conservative part of the system non-integrable in the Liouville
sense, while the electromagnetic self-force causes the charged body to
inspiral. By studying the system without the self-force, we show the existence
of an approximate Carter-like constant and discuss how resonances grow as a
function of the perturbation parameter. Then, we apply the electromagnetic
self-force to investigate crossings of these resonances during an inspiral.
Averaging the energy and angular momentum losses during crossings allows us to
employ an adiabatic approximation for them. We demonstrate that such adiabatic
approximation provides results qualitatively equivalent to the instantaneous
self-force evolution, which indicates that the adiabatic approximation may
describe the resonance crossing with sufficiently accuracy in EMRIs.
