This is an excellent issue for discussion - as it is certainly something I have been thinking about in other cichlids. Sometimes the inbreeding appears to create dwarfism, which I personally don't mind all that much, depending upon the fish, but you wonder what is next down the road. There are certainly a plethora of other less desirable physical traits that crop up, like Madasa mentioned.
I am very curious about whether suceptibility to illness is a part of it, and if so, how long did it take to happen, and is this happening in other, perhaps less popular breeds that have been captive bred for a long time. I know genetic lines are heavily discussed among betta breeders, so there might be something to learn from what they know.