They don't rub together but sometimes one of the inner lips will get twisted and rub against my knickers. If I don't notice straight away it gets sore!
You know, the phrase "bush fire" was coined in the 1970's when a woman with a 'compact' vagina and excessive pubic hair was participating in a 100m sprint. The friction and the dry fuel source created an inferno. Which funnily enough was at the time called a 'Disco Inferno'. Look it up on Wikipedia.
During my years as a runner, I never had problems with my pussy lips rubbing against each other or against my underwear/running shorts. I would guess this is not a problem for most women...I think when most of us run, the lips don't move with the motion of the legs; they stay stationary against each other while the rest of the body moves. Hope that makes sense.