I started walking around WDW at the age of 2 and I'm 20 now. I also have asthma. Walking didn't ruin my vacation either. I bet you and I did the same thing when we needed a rest -- sat down & took a break! There was so much to see and do, here was no way I'd be camping out in a stroller or falling asleep.
PlainJane and I are both living proof that it will not kill your child to walk around WDW if they do not have a disability preventing them from doing so.
If you never try letting your kid walk around, how will you know? Children are being conditioned to not have to walk anywhere. Walking is healthy, even for little bodies. When you get tired, you take a rest. Parents who are forcing their kids to go-go-go non-stop at WDW are just creating cranky and stressed children. Go easy. Walk and take breaks.
My mom's attitude with me was this: if you want to get somewhere, you walk, and if you don't want to walk, then I guess you're not going. If you want to drink, drink out of a cup, (soda) bottle, or can, because the baby bottle is gone and we don't do sippy cups. If you want to sleep, there's the big girls bed, and the crib is gone. If kids have no other choice, they do what they need to do, and adjust. That's real life.
That's my opinion. I'm sure you are all offended. I'm sorry, and I truly do not mean to offend anyone. But this has worked for every child in my family and none of us are screwed up because of it. If you are pushing your nine year old around in a stroller at WDW full-time (not just hopping into a younger sibling's stroller for a little bit), really step back and take a look at that.