The moderate or intense low oxygen dilution (MILD) combustion regime offers reductions in pollutant emissions and improvements in efficiency. The implementation of MILD combustion in non-premixed systems, however, still requires a significantly improved understanding of the effects of the oxidant stream composition in both the MILD combustion regime, and in the transition from MILD combustion to conventional spontaneous-ignition. This paper presents a numerical study of two-dimensional flames near the transition between MILD combustion and conventional spontaneous-ignition. The numerical study is performed with a detailed kinetic mechanism, using the laminarSMOKE code to provide insight into the chemical structure of flames with a hot and diluted coflow. This study is complemented by new experimental observations of laminar flames in similar coflows, which demonstrate a non-monotonic change in lift-off height with changing oxidant O2 concentration. This change occurs in conjunction with a transition from spatially gradual ignition to well-defined flame bases, with increasing coflow O2 level. The simulated flames are compared to a previous definition of MILD combustion in a hot coflow as an edge flame without a tribrachial flame structure. The different structures of simulated CH4 flames are consistent with the observed experimental behaviour under similar conditions, however comparisons between experimental observations and simulations of C2H4 flames highlight the importance of the flow-field, even in a simple streaming flow. The simulations show that equilibrium levels of the OH radical (< 10 ppm) in the oxidant stream significantly intensifies a MILD CH4 reaction zone, by increasing CH3 oxidation, however such levels have little effect on tribrachial, spontaneously-igniting flames. Conversely, increasing the ratio of CO2 to H2O in the coflow reduces the intensity of a MILD CH4 reaction zone. Neither the inclusion of equilibrium concentrations of OH, nor the change in CO2 to H2O ratio, in the oxidant results in a transition between MILD reaction zones to tribrachial spontaneously-igniting flames, despite significantly affecting the temperature of reaction zones in the MILD combustion regime. The results show that the intensity of MILD combustion is strongly dependent on the different chemical species in the oxidant stream. Tribrachial spontaneously-igniting flames are, in contrast, relatively resilient against changes other than temperature and O2 level.