Downward-facing dog might be your favorite yoga pose, but that isn't quite what I mean when I'm talking about the yoga of dog training.
There are dog trainers who have probably never been inside of a yoga studio but whose way with dogs is definitely a yoga. Mike, from whom I picked up Ellie, is one of those dog training natural yogis, although I'm not sure he would think of himself in that way. He just knows dogs, what they need, and how to interact with them. The rest of us need to develop our awareness and observation skills when interacting with our dogs, and that's where learning yoga tools is so helpful.
The yoga tools that I find helpful in my work with dogs and other animals include: asana (postures) and vinyasa (movement) to develop a strong and centered stance (very useful when with a dog that pulls), breathwork (useful for calming and regulating energy levels up and down when needed) and meditation tools for focusing awareness, observing behaviors, and learning more deeply about the dog.
Ellie has a very strong prey drive, and her focus is naturally outward, towards things that move (like cats). My focus, in training, has to be on Ellie, and how to help her direct some of that acute attention towards me. I'm using a combination of clicker training, and neutral techniques called 'park' and 'racetracking'.