Dynamic Sitting: a Feldenkrais® look at posture

One of the most frequent requests I get in my private Functional Integration® lessons is for help with sitting. My students tell me they have trouble finding a comfortable position to sit in, or that they can't make it through a work day or meditation session without discomfort. They have lower back pain or RSI, hand pain, tight neck and shoulders... you get the idea. We all sit a lot, and we all face these challenges.
"I know I have terrible posture," they often say. Or else "I try to sit up straight, but I get tired," or "I keep reminding myself to sit up, but the next time I check, I'm slouching again." Often their idea of "good posture" is to stiffly pull the shoulders back and tighten their belly, which is usually counter-productive. They are coming to me to help them find a better way.
While private FI lessons are well-suited to addressing this sort of question, I thought it would be useful to spend some time exploring the topic of sitting in my classes and workshops. I'm sure it will come as no surprise to you that we will be looking for solutions to the problem that have to do with movement and awareness, rather than the usual approaches of proper alignment, ergonomics, and core strength.
We’re all pretty familiar with this problem:
Or, put another way:

We certainly did not evolve to be sitting creatures, especially not sitting in chairs and staring at a computer screen. Yet so many of us spend so much of our time doing just that, and most of us know that we don’t do it in an optimal way. We don’t need a physical therapist or a mom to tell us that our posture is all wrong, we can feel it in our aching backs and sore shoulders and necks. (That said, we may need a professional to point out to us that our so-called repetitive strain injuries are also linked to the way we work at our computers).

So, what’s the solution?