I think when they're young and have no option but to eat the food that the parent provides/exercise and be active as the parent see fit, then yes of course the unhealthiness will continue.
I was brought up in a VERY unhealthy household/family. Grandmother was morbidly obese, suffered through terrible diabetes and passed as a result of that. My mother and her siblings (bar one) are all very unhealthy and overweight, all have diabetes/sleep apnoea/high blood pressure/gout, do nothing much to discontinue going down the exact same road that their Mother did. My Mum tried hard to keep me healthy but as I got older (8 or so) she got tired and depressed and we lived on microwave dinners/kfc/mcdonalds etc. I always loved healthier food and didn't want to eat meat anymore though so when I was 12 or so I asked to be allowed to do the groceries/cooking. She continued to eat her take away etc but I very rarely did. I didn't want to be like the rest of my family (still don't). It was very hard but I wanted it that bad. Needless to say I'm very healthy and haven't been overweight since I was 14 or so. It's not always easy though, I still crave junk food ALLLL the time, comfort eat etc and I greatly attribute this to my upbringing. It's still hard when I have meals out with my family, it's always eat, eat, eat, pressuring me to have dessert etc. I thank my lucky stars that for whatever reason as a kid I saw that it really wasn't that hard to live healthily and soooo so so worth it. It makes me very sad watching my family, especially my Mum, struggle so much with their health though. She tries, but it's just been way too long and she's too set in her ways.