My experience of how special assistance works is that when you arrive, you announce yourself to the special assistance desk. They usually go over what kind of assistance you need so you can let them know of any specific concerns or requests. Then (assuming you want it) someone will meet you and walk you through check-in and security, and take you to a special assistance waiting area, which will probably be some distance from the gate.
They will leave you in the waiting area until close to boarding time and it’s during this time that your mum is most likely to be able to go walkabouts. They usually give you a time when you have to be back at the waiting area, and many have pagers that they give to you, which beep and flash when it’s time for you to be back.
At the designated time, they will go round the waiting area collecting everyone for a particular flight and take everyone to the gate together. The agent then typically waits at the gate, because boarding is imminent. They may not wait right next to you but usually remain in the vicinity. When boarding starts, they collect up the people they’re responsible for and help them through the gate and as far as the plane door.
We never get help onto the plane itself and if someone has difficulty walking, I’m not sure if it’s the airline or the airport’s responsibility to get them onto the plane.
Since it’s the individual airport that provides the special assistance (even though it has to be booked through your airline), I have found the airlines know very little about the details of how anything is done. You can, however, call the airport itself and ask to speak to someone in special assistance (or contact them via an online form if they have it available). In your case, I would do this for each airport your mum will transit through.
I should mention that this is what I've experienced in European airports (Copenhagen, Amsterdam, Heathrow, Stockholm, Inverness) and all airports are different, so it might operate differently elsewhere.
ETA: as per the comments, no, I haven't addressed every variable, because there are as many variables as there are people who need assistance, assistance agents and assistance companies. Maybe we've been lucky with flight timings; maybe it's because I've been travelling too and therefore able to communicate our needs clearly at all stages; maybe it's because the person I accompany visibly looks like they shouldn't be left alone. The fact is, people receiving assistance are not prisoners and agents are not keepers - anyone can wander off at any point during the process.
The person I travel with wears a sunflower lanyard that indicates hidden disabilities. I'm not sure it's universally recognised yet, but awareness is growing and it alerts airport staff (and other passengers) that he might need help, even when it's not blindingly obvious. If your mum's not aware/alert enough to consistently communicate her own needs then you might consider getting a lanyard for her.
