  If dark matter is composed of new particles, these may become captured after
scattering with nuclei in the Sun, thermalise through additional scattering,
and finally annihilate into neutrinos that can be detected on Earth. If dark
matter scatters inelastically into a slightly heavier ($\mathcal{O} (10-100)$
keV) state it is unclear whether thermalisation occurs. One issue is that
up-scattering from the lower mass state may be kinematically forbidden, at
which point the thermalisation process effectively stops. A larger evaporation
rate is also expected due to down-scattering. In this work, we perform a
numerical simulation of the capture and thermalisation process in order to
study the evolution of the dark matter distribution. We then calculate and
compare the annihilation rate with that of the often assumed Maxwell--Boltzmann
distribution. We also check if equilibrium between capture and annihilation is
reached and find that this assumption definitely breaks down in a part of the
explored parameter space. We also find that evaporation induced by
down-scattering is not effective in reducing the total dark matter abundance.
