Not much time to post as most of this last week and the next two are spent in hospital (fortunately as a job & not a patient!), but a quick off the top of my head summary-
Halons were originally thought to smother in the days of CTC, but it was soon realised they inhibited the combustion process. Different Halons had different 'inhibitory factors', the lower the factor the lower the concentration of halon required to interrupt combustion. Methyl Bromide is the most effective, being 3 (only 3% concentration required) but was far too toxic. Their inhibitory effect is by their affinity for 'free radicals' produced by combustion -by binding with the agent they cannot propagate the react further. Powder works in a similar way.
Inert gases are just that - they work by displacement of oxygen, their inert nature making them unable to contribute to the combustion process.
Chemical agents have the advantage of being more effective mass for mass due to their chemical interference, hence why a 2.5 kilo BCF extinguisher had the same fire rating as a 5 kilo CO2 extinguisher, but are resultantly more expensive, hence why you could fill a 2 kilo CO2 extinguisher for around £2, but a 2.5 kilo BCF cost £20+