A few years back, I was paired up as the "champion" of my office against the "champion" of another office (we are all paralegals in the Army, and competition is very active for ***** talking rights anywhere we can earn them).
I am a kung fu player, internal arts primarily with little other experience at that time (since I have studied Modern Arnis and Ryu Te Karate). He was a wrestler trained in Iowa by a group of brothers that are apparently very well known for training collegiate champions (I can't remember their names right now, but I'd know them if I heard them).
Anyway.
We started circling each other, and in short order my opponent took me to the ground. I went onto my back immediately, which took him by surprise (he had expected me to keep him from pinning me, but being pinned was the least of my concerns). I began alternatingly freeing my hands, reaching out for vital points, seizing points, etc. I didn't strike him with any force, just lightly touching him repeatedly (we are fellow soldiers, and getting into a real fight is not only a real No Go, but we were friends to boot...).
His side kept pointing out how well he was keeping me pinned beneath him, unable to get him off of me. My side pointed out to his side that every time I was touching him was one strike that he failed to deal with...
When I trained in Modern Arnis in Japan, my instructor was also a big grappling fan. I will admit, were he and I to get into a fight it would get really ugly, and I have absolutely no idea who would win... I would like to say I would, but I know better... The sombo and BJJ stuff he showed me was very nasty, and I know that grapplers do not refrain from striking their downed opponent. However, I believed then, and do so now, that traditional CMA, and traditional MA for that matter, have sufficient skills present in their curricula to allow for dealing with grappling situations.
As noted on some other threads, it all amounts to how you train.
Make friends with grapplers. Invite them to your school or training hall, and play around with some situations. See if you can find ways to apply what you have been taught to that kind of situation. Only then will you really know.
Gambarimasu.