I was asked to write something here too, as most of you know, I am a certified centered riding instructor.
I did the course to have more understanding of the influence of the rider on the horse from a more biological and passive point of view.
I have to say though, that I only use a few things of centered riding. The breathing low for instance. But I used that already from my meditating back ground.
Next to that I use certain other things as indeed mentioned here, the swinging barrel, but also creating the feeling like your feet are on roller skates is an excellent one.
I think that is is a very good thing to focus on everything centered riding has to offer for a while to get to know one's body and the influence of the body on the horse.
But... (is there not always one?)... after a while, in my experience, it is better to simplefy things back to 'just riding'. Or else one is going to mis a lot of what is happening with his horse all the time.
Because, well let's face it, eventhough riding is a lot of working on the rider, the goal is to train the horse.
So, after I work on the rider, I focus back on the horse, on his
training, telling the rider what to do from reading the reaction of the horse.
By trial and error the rider learns what to do and what not. I let the rider make mistakes on purpose and then tell him what to do, and FEEL the difference. You need to feel 'what have I done, and what was my horse's reaction'.
And it won't work thinking about keeping your toes in for instance, if you know what I mean.
The key ingredient in this is bitless/loose rein/cordeo riding.
For only then can one get a clear and true answer of the horse, in my view.
So, in conclusion, centered riding is one of the best ways to learn how to 'be on a (moving) horse'.
When you get the basics, you have to start actually training though.
When the horse rewards certain behaviour of you, you will learn what do to.
But you have to learn first, what the 'reward' feels like
If any questions, shoot!