The electron kinetics in nanowire-based hot-carrier solar cells is studied, where both relaxation and extraction are considered concurrently. Our kinetics is formulated in the many-particle basis of the interacting system. Detailed comparison with simplified calculations based on product states shows that this includes the Coulomb interaction both in lowest and higher orders. While relaxation rates of 1 ps are obtained, if lowest order processes are available, timescales of tens of ps arise if these are not allowed for particular designs and initial conditions. Based on these calculations we quantify the second order effects and discuss the extraction efficiency, which remains low unless an energy filter by resonant tunnelling is applied.