We have traveled
many, many times with DD and her wheelchair.
What we do (and what you want to do) is called "Gate Checking" the wheelchair.
You should tell the first person you check in with that you want to gate check the wheelchair. They may give you a special tag and instructions or tell you to talk to the gate agent. You want to do that before boarding starts, so that the tag can be filled out correctly.
When you gate check, you keep the wheelchair until you actually board the plane. You can use it to get down to the door of the plane (and depending on just how things are set up) can usually actually wheel it into the entrance of the plane. After taking your child out, one person in your party can get her seated whil;e the other takes the wheelchair apart and folds it. Remove anything that comes off easily or could pull off during transit (seat cushion, seat back, armrests and footrests if they are swing away ones and could be pulled out if they are pulled on). We sometimes take a large laundry bag to keep the pieces contained; carry the pieces on and put them in the overhead bin or plane closet.
When you gate check the wheelchair, you can request it be brought back to the gate at your destination city
or your connecting city or to baggage claim. Make sure the tag is filled out correctly. The last time we flew, we saw someone who gate checked the wheelchair, but they apparently just thought you could do this without discussing it with anyone. Their wheelchair did not have a gate check tag and ended up on the baggage cart going to baggage claim. We could see this out our window and when we went to get DD's wheelchair (which was correctly marked) could hear them complaining to the FA that they had left it at the gate when they got on and expected it to be there when they got off, but didn't know you had to do anything besides leave it there.
Occasionally, there will not be enough time to get it up to the gate at your connecting city and back down again at the gate when you leave. But, that's occassional and the gate agent will be able to help you with that (plus your
travel agent or a call to the special needs department of the airline before you go should let you know).
An alternative is to
not gate check the wheelchair, but bring it on board with you. Most flights have a closet that a wheelchair
could fit in, especially if you can pop off the tires and fold the wheelchair. Some flights don't and if they have one, but someone requested it before you, they don't have to accomidate you. So, ask about this when making your arrangements before you leave.
The flight attendants will not usually be helpful (or not that helpful), especially the FA for First Class, because he/she may have to explain to his/her passengers that the "First Class" closet is not available for their use. Wheelchairs/mobility devices take precidence over luggage. The gate agent usually knows the rules and will be your friend. (Can you tell we have done this particular fight several times?)