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We present the results of dust continuum and molecular line observations of two massive cluster-forming clumps, NGC 2264-C and NGC 2264-D, including extensive mapping performed with the MAMBO bolometer array and the HERA heterodyne array on the IRAM 30m telescope. Both NGC 2264 clumps are located in the Mon OB1 giant molecular cloud complex, adjacent to one another. Twelve and fifteen compact millimeter continuum sources (i.e. MMSs) were identified in clumps C and D, respectively. These MMSs have larger sizes and masses than the millimeter continuum condensations detected in well-known nearby protoclusters such as ρ Ophiuchi. The MMSs of NGC 2264 are closer in size to the DCO⁺ ``cores'' of ρ Oph, although they are somewhat denser and exhibit broader linewidths. Most of the MMSs of NGC 2264-C harbor candidate Class 0 protostars associated with shocked molecular hydrogen jets. Evidence of widespread infall motions was found in, e.g., HCO⁺(3-2) or CS(3-2) in both NGC 2264-C and NGC 2264-D. A sharp velocity discontinuity ∼2km/s in amplitude was observed in N₂H⁺(1-0) and H¹³CO⁺(1-0) in the central, innermost part of NGC 2264-C, which we interpreted as the signature of a strong dynamical interaction between two MMSs and their possible merging with the central MMS C-MM3. Radiative transfer modelling supports the idea that NGC 2264-C is a highly unstable prolate clump in the process of collapsing along its long axis on a near free-fall dynamical timescale ∼1.7x10 ⁵yr. Our model fit of this large-scale collapse suggests a maximum mass inflow rate ∼3x10^–3M_☉/yr toward the central protostellar object C-MM3. In NGC 2264-D, we estimated a mass infall rate {dot}(M)_DMM1∼1.1x10^–4M_☉/yr toward the rotating Class 0 object D-MM1, also based on radiative transfer modelling of the observations. Such infall rates are sufficiently high to overcome radiation pressure and allow the formation of ∼20M_☉ stars by accretion in ∼1.7x10⁵yr, i.e., a time that is similar to the global dynamical timescale of the central part of NGC 2264-C. We conclude that we are likely witnessing the formation of a high-mass (>10M_☉) protostar in the central part of NGC 2264-C. Our results suggest a picture of massive star formation intermediate between the scenario of stellar mergers of Bonnell et al. (1998MNRAS.298...93B) and the massive turbulent core model of McKee & Tan (2003ApJ...585..850M), whereby a turbulent, massive ultra-dense core is formed by the gravitational merger of two or more Class 0 protostellar cores at the center of a collapsing protocluster.