Both Co3O4 and LiCoO2 have the same FCC oxygen sublattice. So the reaction takes place mainly via diffusion of Li into Co3O4 and rearrangement of cations (all tetrahedral positions become vacant and Li/Co ordering appears).
My personal opinion after analysis of the HT XRD data is that some intermediate spinel phase (approximately LiCo2O4) is formed initially and only after concentration of Li becomes too high the cations start moving from tetrahedra to octahedra.