At low thermal power (<5 kW) conditions, nitrogen and carbon dioxide were added as diluents to a premix of methane-oxygen in an atmospheric generic swirl burner. Results indicate that CO2-diluted oxy-methane flames have a wider stability range than N2-diluted flames in terms of overall oxygen concentration in the premix. Bulk flow Reynolds number, augmented by varying the size of the burner exit nozzle, was also found to increase the stability limits of flames diluted with both CO2 and N2, as the increased flow velocity offsets the higher burning velocity of the oxyfuel mixture. A combination of differing transport properties between diluents and the resulting flame chemistry produces a change in the structure of the premixed oxyfuel swirl flame, shown by combustion PIV to affect the observed lean and rich stability limits. Utilising the results at low thermal power conditions, enhanced-oxygen combustion of a methane-air flame was investigated in a pressurized generic swirl burner operating at higher thermal power (<50 kW) conditions and pressures up to 3 bar absolute. Over a range of increasing thermal powers, it is seen that a relatively small amount of pure oxygen addition can shift the equivalence ratio at which the lean stability limit or rich stability limit are reached compared with the same phenomenon observed for a methane-air flame. Pressurised operation with CO2 dilution up to 15.5 mol% was validated through stability limit and emissions gas analysis, giving further support to the use of exhaust gas recirculation in premixed swirl-stabilized burners for oxyfuel combustion.