Most strollers are too big to take as carry on.
Strollers are always gate checked. You wheel the stroller down the jetway, take your kids out, and leave it by the door. After everyone has boarded, either the gate agent or the ramp gate crew will take the strollers and any carry-on bags that didn't fit in the overhead bins and stowes them in either the bulk bin(for widebody aircraft) or the front compartment (generally bin 1) in the belly of the plane (this bin is usually the last one filled up in an aircraft). When the aircraft lands that bin is the first one open and the gate crew takes the strollers and leaves them by the door of the plane. The gate checked bags end up at baggage claim or transferred to a connecting flight. But if you are flying with a stroller, you really don't need to bring your every day stroller that seats 4, has 6 cup holders, and has enough storage space you could live out of it for a month on your 5-day vacation to wherever. Those things are a bitch to try to lug up those stairs, especially when you have 4 of them. The small ones that fold up (some even have carrying bags with a shoulder strap) are so much easier and lighter. I have worked as a gate agent and on the ramp on gate crews at the busiest airport in the world, and my advice is this: make things as easy as you can for the crews, and your things are less likely to get damaged. Not out of malice, but because the amount of gate checks gets larger every year and right before departure is the busiest time for the gate crew.