The capability of radioluminescence (RL) dosimeters composed of carbon-doped aluminium oxide (Al2O3:C) detectors+optical fibre has been verified for absorbed dose-rate measurements during carbon radiotherapy. The RL signals from two separate Al2O3:C detectors (single crystal 'CG' and droplet 'P1') have been systematically measured and compared along the Bragg-curve and Spread-Out Bragg-Peak of 290MeV/n carbon beams in the water. The absorbed dose response was assessed for the range of 0.5-10Gy. For doses up to 6Gy, we observed a linear response for both types of detectors, while for higher doses CG presented a more prominent supraliearity than P1. The RL response for low-LET protons in the entrance from the curve was found to closely resemble that observed for a clinical 6MV X-ray beam, while it was found that P1 has a better agreement with the reference data from standard ionization chamber than CG. We observed a significant decrease in luminescence efficiency with LET in the Bragg peak region. The Al2O3:C RL luminescence efficiency differs from Al2O3:C OSL results, which implies that the signal can be corrected for LET dependency to match the correct SOBP and Bragg Peak.