Nauli Kriya is a strange and cool looking exercise that makes it look like you are churning your belly. While strange to look at, it's not that difficult to learn. It is also a great exercise for stretching your diaphragm, exercising your intercostals and developing abdominal control. It's also good for warming up.
Note that you shouldn't do this exercise on a full stomach. Empty is best. Also avoid it if you have a hiatus hernia (acid reflux) or "regular" hernia.
The first step is to learn how to do uddiyana bandha. While standing with your knees bent and hands on your thighs with arms straight, exhale all of your air. While holding your breath push down on your thighs with your hands and expand your ribcage as if trying to inhale.
Normally when you expand your ribcage it causes you to inhale because you are expanding the volume of your lungs. However if you lock your throat and prevent the ingress of air as you expand your ribcage, you create a vacuum. That vacuum sucks your diaphragm up into your ribcage. Since your stomach, liver, kidneys and most other organs within your abdominal cavity are suspended from the bottom of your diaphragm, they get pulled up with it. Your belly hollows out as a result.
Pushing down with your hands on your thighs helps you to pull your ribcage up.
To release, first relax your ribcage, then take an inhale.
Rest for a moment (particularly when just learning) and repeat when you are ready.
To make your belly extra hollow while doing both uddiyana bandha and nauli kriya, suck up on your diaphragm. Do this as you lift and expand your ribcage. When it comes time to release, relax both your ribcage and your diaphragm and then take an inhale.
The next stage of learning nauli kriya is to practice "popping" your rectus abdominus.
Again, exhale all your air, then press your hands down against your thighs and lift your ribcage. Press your hands down against your legs to keep your ribcage pulled up. Then engage your abs as if to pull your sternum towards your pubic bone. But resist with your hands.
If you are doing it right your rectus abdominus muscle will pop forwards. However the sides of your belly will be sucked in behind because of the vacuum you've created.
Release your rectus and relax for a few breaths (or more.) Then do the same thing again.
To get your rectus to move in a churning action, swirl your ribcage from side to side. To make it look like your rectus abdominus is "churning" try moving your ribcage in a horizontal circle over your pelvis. Move it to the side, then back, then to the other side then forwards again. To keep your rectus popping, keep holding your breath out and keep your ribcage lifted by pressing your hands into your thighs.
To release, first relax your belly and then your ribcage, and then slowly inhale. You have to unlock your body otherwise the inhale can be uncomfortable.
It's all in the video.
As simple breathing exercise is to focus on pulling your ribs up, away from your pelvis each time you inhale. Then as you exhale relax. Work at slowly expanding and lifting your ribcage as you inhale. Work at relaxing slowly as you exhale.
This particular method of breathing uses the (external) intercostals to expand the ribcage The intercostal are the muscles situated between the ribs.
Once you've got the hang of inhaling and exhaling smoothly then work at exhaling completely.
For this exercise focus on squeezing your abdominals and ribcage inwards as you exhale. You might try a few pushes near the end to wring every last drop of air out and then try to smoothly inhale afterwards.
To get even more air out you can curve your spine forwards as you exhale. You may find this happens naturally the more you use your abdominals.
You can also practice relaxing your belly at the end of your exhale while holding your breath.
First exhale using your abs to help. Hold your breath and relax your bellly. Then try inhaling smoothly. You may find that your breath catches slightly. Next try the same thing, exhale completely, relax your belly, then reengage your belly and then inhale. You may find that the start of your inhale is smoother.
The next phase in nauli kriya training is to exhale all of your air, then relax your belly, and then expand and lift your ribcage without inhaling. You may find your throat feels like it is being sucked inwards as you do this.
To inhale, first relax your ribcage, then pull your belly in and then inhale.
When practicing nauli kriya work towards three repetitions, holding the exhale for as long as comfortable. Rest after each repetition. You can also work at gradually increasing the number of times you churn your belly. Try churning in both directions. See if you can gradually work towards ten in one direction and ten in the other all on one breath hold.
Why do nauli kriya? Its a good way to give your intestines a helping hand first thing in the morning. It's energizing and thus a good way to warm up prior to a yoga practice. It stretches the diaphragm and it exercises the intercostals as well as the rectus abdominus. It also stretches the obliques, pulling the inwards during the vacuum phase.
As a yoga teacher, I'm constantly exploring new exercises, new ways of doing yoga poses.
There is no single "right way" of doing a yoga pose. Instead, there are options. And the better you are at "feeling" your body, the better you can get at choosing the right option for your body as it is now.
For any technique, the point of practice is to learn feel it and to control it, so that it can be used without thinking about how to use it.
And that is more or less the approach taken in all of my ebooks and videos. They help you to feel your body and control it so that you can work towards using it effectively in anything that you do.