Even when pursuing semi-tame animals at a petting zoo, chasing deer requires an intensity and dedication that most do not possess. As best I can tell, a deer is able to lightly scamper away while the human chaser exerts at a level that quickly leads to involuntary upchuck or to a severe side stitch or both. But I imagine some of the less strict parents and their misbehaving children need this warning, both for their safety and the safety of the scampering deer.
Parents, you are indeed welcome to provide leadership to your children. And to avoid confusion and maximize absorption, provide said leadership with reduced speed. We wouldn't want to overwhelm the children. And if you want to walk and lead the pony that your child is riding, you're welcome to do that too. But that's something totally different, and unrelated to leadership of children.