It depends a lot on the nature of the lessons and how they are paid for. For one thing, group classes are different from private lessons.
I have at times been blessed with a teacher who doesn't ask for money. In that case, I have a hard time saying the lesson is running overtime / undertime. We make arrangements on when/where to meet, and for how long. If it's tight in the schedule, we say so and try to stick to the agreed to times.
In general for paid lessons:
If we're running late, I expect the lesson to end on time. If the instructor is running late, I expect the lesson to go to the allotted end. If a teacher/instructor is willing to give my child extra attention / instruction after the lesson (positive/focused on the subject), I am grateful.
We also try not to keep a tight schedule. DS is an only so far, which helps with this.
If a class has a history of going long, and for some reason we have a tight schedule that day, I expect I would say something to the instructor at the beginning about needing to leave at x time. And then (if the lesson isn't over), I'd step in at that time, make eye contact with the instructor, and expect DS to be dismissed. If I have to say something, it would be like "I'm sorry, [DS] needs to go now."