It all sounds so terribly simple, doesn't it? Take the stroller and tell the passenger it will be there for her when she arrives at her destination.
Well how about this:
My wife, my infant sprog and myself were on a BA flight from Frankfurt to Heathrow. I don't remember what the aircraft was, probably a 737. Whatever, I was sat next to the window and I looked out out of my window to see our stroller (push chair to me!) and the baby's car seat being brought out to the aircraft for loading. The person bringing them disappeared beneath the aircraft immediately below where I was sat, presumably for stuffing into a hold. Dead simple.
Then we arrived at Heathrow to discover the push chair and baby car seat had failed to make it out of Frankfurt! A fair old bit of "Oh dear, we are so sorry" etc but no suggestions as to how I was going to hire a car to take the family home when the baby's car seat was hundreds of miles away in Frankfurt! It ended up with a taxi home and no compensation.