This post applies specifically to Kawasaki Vulcan with a Keihin CV carb, but is the same as a Harley Keihin CV carb.
When your carb is nasty all over the accelerator pump and no where else on the body and you notice a bit of a roll-on stumble, you need a new accelerator pump diaphragm.
Pull the carb and turn it over on the bench. These three screws hold the pump together. Remember to watch for Japanese screws.
There it is.
There's the tear. Swap it out. A HD diaphragm is the same as Vulcan or an S&S Super E.
I had to bench check it. I have bottle of rubbing alcohol I use in my auxiliary tank bottle to
test bench test float heights, but I used it to check the accelerator
pump function after replacing the diaphragm.