Sketch depicting an example orbit change following our orbit-averaged procedure. The stellar-mass BH, initially on the orbit labelled 0, is kicked when r = a, and ends on the orbit labelled 1. Both orbits have the same semi-major axis a, but different eccentricities. The dashed circle represents a circular orbit with radius a. In this example there is no way to rotate either orbit around the focus occupied by the MBH such that the pericentre or apocentre of orbit 0 crosses orbit 1 at any point. Thus, neither pericentres nor apocentres are good spots to employ our orbit-averaged procedure. We note that the eccentricities of these example orbits are much lower compared to the ones investigated in this work. Moreover, the difference between subsequent orbits is much exaggerated here.