The role played by protostellar feedback in clustered star formation is still a matter of debate. In particular, protostellar outflows have been proposed as a source of turbulence in cluster-forming clumps, which may provide support against global collapse for several free-fall times. Here, we seek to test the above hypothesis in the case of the well-documented NGC 2264-C protocluster, by quantifying the amount of turbulence and support injected in the surrounding medium by protostellar outflows. Using the HERA heterodyne array on the IRAM 30m telescope, we carried out an extensive mapping of NGC 2264-C in the three molecular line transitions ^12^CO(2-1), ^13^CO(2-1), and C^18^O(2-1). We found widespread high-velocity 12 CO emission, testifying to the presence of eleven outflow lobes, closely linked to the compact millimeter continuum sources previously detected in the protocluster. We carried out a detailed analysis of the dynamical parameters of these outflows, including a quantitative evaluation of the overall momentum flux injected in the cluster-forming clump. These dynamical parameters were compared to the gravitational and turbulent properties of the clump. We show that the population of protostellar outflows identified in NGC 2264-C is likely to contribute a significant fraction of the observed turbulence but cannot efficiently support the protocluster against global collapse. Gravity appears to largely dominate the dynamics of the NGC 2264-C clump at the present time ; however it is possible that an increase in the star formation rate during the later evolution of the protocluster will trigger enough outflows to finally halt the contraction of the cloud.