Aims. We aim to characterise certain physical properties of high-mass star-forming sites in the NGC 6334 molecular cloud, such as the core mass function (CMF), spatial distribution of cores, and mass segregation. Methods. We used the Atacama Large Millimeter/sub-millimeter Array (ALMA) to image the embedded clusters NGC 6334-I and NGC 6334-I(N) in the continuum emission at 87.6 GHz. We achieved a spatial resolution of 1300 au, enough to resolve different compact cores and fragments, and to study the properties of the clusters. Results. We detected 142 compact sources distributed over the whole surveyed area. The ALMA compact sources are clustered in different regions. We used different machine-learning algorithms to identify four main clusters: NGC 6334-I, NGC 6334-I(N), NGC 6334-I(NW), and NGC 6334-E. The typical separations between cluster members range from 4000 au to 12 000 au. These separations, together with the core masses (0.1–100 M_⊙), are in agreement with the fragmentation being controlled by turbulence at scales of 0.1 pc. We find that the CMFs show an apparent excess of high-mass cores compared to the stellar initial mass function. We evaluated the effects of temperature and unresolved multiplicity on the derived slope of the CMF. Based on this, we conclude that the excess of high-mass cores might be spurious and due to inaccurate temperature determinations and/or resolution limitations. We searched for evidence of mass segregation in the clusters and we find that clusters NGC 6334-I and NGC 6334-I(N) show hints of segregation with the most massive cores located in the centre of the clusters. Conclusions. We searched for correlations between the physical properties of the four embedded clusters and their evolutionary stage (based on the presence of H II regions and infrared sources). NGC 6334-E appears as the most evolved cluster, already harbouring a well-developed H II region. NGC 6334-I is the second-most evolved cluster with an ultra-compact H II region. NGC 6334-I(N) contains the largest population of dust cores distributed in two filamentary structures and no dominant H II region. Finally, NGC 6334-I(NW) is a cluster of mainly low-mass dust cores with no clear signs of massive cores or H II regions. We find a larger separation between cluster members in the more evolved clusters favouring the role of gas expulsion and stellar ejection with evolution. The mass segregation, seen in the NGC 6334-I and NGC 6334-I(N) clusters, suggests a primordial origin for NGC 6334-I(N). In contrast, the segregation in NGC 6334-I might be due to dynamical effects. Finally, the lack of massive cores in the most evolved cluster suggests that the gas reservoir is already exhausted, while the less evolved clusters still have a large gas reservoir along with the presence of massive cores. In general, the fragmentation process of NGC 6334 at large scales (from filament to clump, i.e. at about 1 pc) is likely governed by turbulent pressure, while at smaller scales (scale of cores and sub-fragments, i.e. a few hundred au) thermal pressure starts to be more significant.