As long as both parks are open, you can go from park to park. I hated it though when they closed DCA, and we stayed late at DL, and then had to get the kid out of the stroller inside of DL. It is much better when you can abandon the stroller at the security gate, as it is that much less carrying.
We took a stroller since we really needed a baggage cart to get us back and forth from the hotel. We used to get the disney stroller for free with the disney visa but since they changed to a discount from free, we just used our stroller and grabbed a "turned in" stroller towards the end of the day for the 2nd youngest. We usually found a stroller with nametag removed at Cal Adv entrance to GCH and helped disney get it to the gates. I've heard you can also check in your stroller when you get the disney stroller so you have something to push sleppy head home in if you are in walking distance.