The authors have studies in this paper the competition between (projectile) core-conserving and core-varying single-electron capture processes in Ne⁶⁺-He collisions for impact energies between 0.07 and 1.2 keV amu^-1. By deconvoluting the spectra of the VUV radiation emitted by the decaying product ions, absolute state-selective cross sections for these processes were determined. The authors also obtained capture cross sections for metastable Ne⁶⁺ ions colliding on helium. Core-conserving capture channels dominate the charge transfer process at the higher impact energies investigated; with decreasing impact energies, however, the core-varying channels gain strength at the expense of the core-conserving channels. The results show that the capture cross sections for the core-conserving processes involving metastable primary ions are comparable with those for the ground-state primary ions. At lower impact energies, however, the capture cross sections for the metastables are almost three times larger than the cross sections for ground-state primary ions.