Don't put a DME tag on the stroller.
When you check in, tell them you want to gate check your stroller. They may give you a gate check tag right then, or (more commonly) they tell you to get one from the agent at the gate. Do that before boarding begins (it is too busy after and there is a chance that a mistake might be made filling it out, which could cause your stroller to go on vacation in another city without you). In some cases, you can have a choice of where the stroller will be delivered - to the gate or to baggage claim of your connecting city or final destination.
You can keep your stroller with you to wheel down the jetway (the tunnel already mentioned). When you get to the door of the plane, fold your stroller up and make sure it is secured so that it doesn't unfold. If there is anything removable (like attached cupholders, canopies, etc.) remove them and bring them on the plane with you.
Folding the stroller and removing things is very important to prevent damage; if you don't fold it, the baggage handlers will and they may not know that (for example) the canopy has to be folded before the rest of the stroller, or that there is a lock that needs to be released before you can fold it. They may damage it while folding it or if it doesn't stay securely folded, it may catch on other luggage and get bent.
It does take a few minutes for the gate checked items to be brought up, so don't be in a big hurry to get out of the plane. Also, remember to pick it up before you leave the plane. This may seem obvious, but we are often the last ones off the plane and we often see a stroller or 2 sitting at the gate, waiting to be claimed by people who entirely forgot they gate checked them. The DH of a friend used to be a flight attendant and he said that happened at least once a week on flights he flew.
