Bev, it usually isn't the airport that determines where the stroller ends up, it's the gate-check tag, at least on US domestic flights.
Gate-checking handles two functions; storing personal mobility devices during flight, and also disposing of the overflow carryon luggage that will not fit in the overhead bins or under the seat. The tags reflect this, and there will either be a system of two different colours of tag, or there will be a place on the tag to mark the delivery location of the item. In many cases, if the tag is not the proper colour or specifically marked "DELIVER TO GATE", it may be sent to the carousels along with the baggage.
Most US airlines use pink tags for mobility devices, including strollers, while they may use yellow for overflow luggage, but there is no guarantee that the tags will always be a predictable colour. You should always first ask the gate agent for a gate-check tag for a stroller, rather than the ramp agent. If the gate agent does not have them, then the ramp agent will handle it, but the gate agent usually has more time to fill the info out properly. Look at the tag after it is put on, and make sure that the information is correct.