Did it just start with the backfiring all at once or has it been been an increasing problem? If it started all at once I'd check the carb float levels and jets to see if they are all as they should be.
Backfiring or after firing is usually caused by the mixture being too lean. I did have a backfiring problem one time with the old shovel when the points collapsed and the spark got real weak. It would shoot blue fire out of it about 5 feet!

Went through 3 different carbs trying to solve the problem before I decided to check the points. I'd just put a new set in and they didn't have 50 miles on 'em when they crapped out.
Pull the plugs and see what they look like. Dry, brown or gray, and it is definitely on the lean side.
Black, sooty plugs that look like they've got a coating of black velvet on them and she's rich. An air cooled engine should always be run a tad rich in my opinion as the excess fuel gets rid of the heat inside the chamber.
Running them lean makes them HOT and will burn up valves and put holes in pistons.
I used to run my old hot rod shovelhead so rich that it would stumble and fart if you pulled the trigger hard on a day with the temps above 80°F. It never turned the exhausts blue or brown though. And it would smoke the tire in 3rd gear at 80mph if you dumped the clutch and twisted the wick coming out of second
BTW, another way to check the cyl temps is to see if the exhausts are blue up near the head, if so, then she's leaner than I like to see 'em.
Let me know what you find.
