Congratulations on solving your FTE problem!
There's probably some way you could spray powder solvent into the assembly and flush out the crud, but for maximum effectiveness, you're better off spending the 5-or-so additional minutes removing the extractor and cleaning it separately. I do this at every *other* cleaning session (I know, I know... I'm lazy), and admittedly, there's very little crud to remove. Using a clean-burning powder (in this case, Hodgdon's Universal Clay) probably helps keep it clean.
DL