On the Safari, there will be a stroller parking area about halfway through the queue. You will have to carry your child or have them walk the rest of the way. On the way out, you'll be able to retrieve your stroller from about the same place. So you will have to either carry your child back out or have them walk. Note that if you decide to the the Pangani Trail right after the Safari (it's easy since you exit from the Safari close to the "entrance"), you won't have your stroller at that point.
For the Wildlife Express, the train takes you out to Rafiki's Planet Watch and Conservation Station. We were just on this over Labor Day weekend so I remember the stroller instructions. All strollers have to be folded up on the train. You will have to fold it up before you cross the yellow line into the waiting area. And you will have to wait at the other end until you are cleared of that yellow line to unfold it. So, you will have to carry your child or have them walk on and off the train and for short distances before you board.