It varies based on airport. I fractured my foot in October, and had an incredibly short connection in LHR in December. It would've been challenging to make when 100% recovered, and I knew I wouldn't have the time given that I was 2 weeks into PT and not walking well. In LHR, I was met at the gate, had all of the expedited services, including a separate shuttle to the plane, with a portion of the "shuttle" bus lifting up to the plane, so I avoided all steps, etc.
At JFK, I again was met at the gate, zoomed through CBP, and out to the passenger pick-up area.
At CLT, there was one employee at the gate, with four passengers requesting services. He gave me a ton of attitude and was not happy to be at work. Given that I saw no option for how he would manage to wheel four wheelchairs at once, I opted to try to hoof it, knowing that I'd likely need to stop and rest.