Bound-state formation can have a large impact on the dynamics of dark matter freeze-out in the early Universe, in particular, for colored coannihilators. For small relative mass splittings between dark matter and the coannihilator, freeze-out can become a prolonged process increasing the relevance of excited bound states. We present a general formalism to include an arbitrary number of excitations in terms of an effective annihilation cross section, taking into account bound-state formation, decay and transition rates. For a coannihilator in the fundamental representation of SU(3)c, we discuss radiative bound-state formation for arbitrary principal and angular quantum numbers n,l as well as electromagnetic transition rates among them. We then assess the impact of bound states within a model with Majorana dark matter and a colored scalar t-channel mediator. We consider the regime of coannihilation as well as conversion-driven freeze-out (or coscattering), where the relic abundance is set by the freeze-out of conversion processes. We find that the latter region is considerably enhanced due to bound-state effects with far-reaching implications for search strategies. In the conversion-driven freeze-out regime, dark matter is very weakly coupled, evading direct and indirect detection constraints, but leading to prominent signatures of long-lived particles that can be probed by dedicated searches at the upcoming LHC runs.