I say DEFINITELY bring your own stroller... preferably something a little better than an umbrella stroller so that 1) it has a sufficient sun shade to shield your little princess from the Central Florida sun 2) it has a decent under carriage basket in which you can stash your extra gear (water bottles, snacks, camera) which otherwise you'll have to continually cart through the parks on your person. 3) most umbrella strolls start to get a bit small for a 4 year old...you want something that provides a decent foot rest and is comfortable enough for snoozing...she will almost definitely fall asleep in the stroller at some point during your trip.
We have three children, ages 2, 4 and 6... All three will be in a stroller for our upcoming trip--our six year old is savy enough to know that if he is willing to sit in a stroller 1) he'll get to more rides more quickly 2) he won't be as tired as his parents who are pushing him 3) his feet won't hurt 4) he won't have to worry about getting lost while he's looking here, there, and everywhere.
THe nicest thing about having your own stroller is that you get to take it with you when you leave the park... This is important when you have a sleeping child and have to get her from the stroller return to the bus which is located a quarter mile away. And it is also helpful when you have to get said child from the bus stop at your resort back to your hotel room a third of a mile away. When carrying your sleeping four year old, you suddenly realize they are no longer babies because they now represent a 38 pound dead weight in your arms! ANd they are awkward as hell to carry while you nurse a limp cause by a persistent blister on your big toe...and as hot humid air causes the sweat to drip off your brow and into your tired blood shot eyes.
We take a double stroller with dual under seat baskets that hold a lot of gear (rain ponchos, sunblock, hats, umbrella, camera, water bottles, bathing suits, a towel). We also take a single umbrella stroller.
Nobody will purposely steal your carriage--whether it be a personal carriage or a Disney rental. Cast Members may MOVE your carriage from where you parked it to keep things orderly outside of attractions...
We ALWAYS tie a colorful bandana on our stroller so we can readily identify it in case there are others that look similar.
If you board an attraction that your Disney rental can not go on (Like the train)...you can get another one at your destination. If someone accidentally takes your rental stroller, likewise. Your own personal stroller can be collapsed and taken on the train with you... (Or you can park it you are taking a round trip on the train).
With a 4 year old you may find you want to occassionally find a strategically chosen parking place for your stroller and then visit nearby attractions on foot...(For instance, in Toon Town). But most places have a stroller parking area specifically designed for this purpose.
HTH?
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