There aren't that many restrictions on breeding sadly. I don't know too much about other dog breeds but people breeding French bulldogs are being encouraged to health test the parents to ensure they're fit to breed from. And some of those criteria are supposed to be looked at in the show ring too but I'm not sure if that's happening in reality. None of it is compulsory though and the huge majority of pups born are to parents who haven't been health tested.If mixed breeds are prone to diseases, why is production continued? For example, some cat and dog breeds were banned for these reasons. The same rule may apply to rabbits.
And, at the other extreme, there are people trying to breed hairless frenchies and all sorts, just because they can and because of the price tag that pups will have.