Composite dark matter is a natural setting for implementing inelastic dark matter - the O(100 keV) mass splitting arises from spin-spin interactions of constituent fermions. In models where the constituents are charged under an axial U(1) gauge symmetry that also couples to the Standard Model quarks, dark matter scatters inelastically off Standard Model nuclei and can explain the DAMA/LIBRA annual modulation signal. This article describes the early Universe cosmology of a minimal implementation of a composite inelastic dark matter model where the dark matter is a meson composed of a light and a heavy quark. The synthesis of the constituent quarks into dark mesons and baryons results in several qualitatively different configurations of the resulting dark matter hadrons depending on the relative mass scales in the system. Show Partial Abstract