Why would misfires occur at rich limits while extreme charge dilutions cause partial burns in gasoline engines?

I'm not sure I understand the boundary conditions for your question correctly, but when you dilute with inert gas, ignition and sustainable combustion will become increasingly difficult. The technical definition of "air/fuel-ratio" will not change due to this dilution since the inert gas is not included in it.

The engine misfires because the gasoline doesn't ignite properly if there is too much fuel and not enough air. It will also produce a lot of unpleasant by products and may fill the exhaust pipe with flammable materials and cause back firing or even explosions. It used to be a common reason for early technology petrol engines to refuse to start and was called 'flooding', and resulted from incorrect use of the choke which cut off the input air and increased the proportion of fuel when the engine was cold.

In a gasoline engine, when rich fuel mixture is used, misfire happens due to lack of oxygen. Theoretical Air/Fuel ratio is 14.7. So, since this ratio is way less during rich fuel mixture combustion, you can experience engine knocking, misfires, and high HC, CO emissions.

Tony Maine and Sudheer Kumar Kuppili are absolutely right, a mixture that is too rich will suffer from misfires. In the literature you'll typical limits for a gasoline engine, e.g. air/fuel ratios (compared to stoichiometric as stated above) between 0.7 and 1.5 are often assumed to be the flammability limit of gasoline. But of course the exact values will depend on details; the range above assumes air as the oxidizer, diluting with inert gases will narrow down the flammability limits.

To be honest I don't understand why it would be desirable to operate an engine at the rich flammability limit and then add EGR. The resulting fuel efficiency and emissions (HC, CO) will be disastrous.

Maybe to offer further insight: since combustion is a reaction between fuel and oxidizer, you could expect symmetrical behavior between the lean and the rich limits. But in reality things are different, i.e. excess air and unburnt/partially burnt fuel will not quench the combustion in exactly the same way.

It could also be helpful to see exactly where the flammability limits are in the experiments that preoccupy you. As I said above, under the right conditions (i.e. type of ignition, turbulence, pressure/temperature, available time...) ignitability / flammability can occur well beyond the "standard" limits.

Thank You everyone for taking the time to provide your insights to my question. All the inputs have been extremely helpful to clear the confusion with the review.