Breath awareness is of course a pivotal part of yoga practice...
... but I think you have to be very careful not to instil the idea that there is 'right' and 'wrong' breathing, because to do so gives people ideas that are extremely restrictive, very open to misinterpretation, and quite unhealthy for the mind/body relationship. However, if you truly understand how breathing mechanisms and postural support can be the same thing, you can include instructions that invite people to work with changing the BODY so that the breath is free to move more appropriately.
You also have to be careful with the idea that you inhale with some movements and exhale with others... This will exaggerate breathing tendencies in a way that might not be helpful. The most commonplace instructions invite a lordotic inhale and and kyphotic exhale, but as you repeatedly exaggerate the curves of the spine, you also get shorter and wider (a.k.a. middle-aged-spread). If I see this going on in a class, I would play devil's advocate and reverse the thing. In my opinion, we are supposed to be free to breathe in AND out whatever we are doing, otherwise we are learning only to be upright when we inhale, and to collapse whenever we let the breath go.
Something that is important to understand about the breath, is that the conditions that allow the body to be supported as breath is released, ARE THE SAME conditions that allow the structure to be open enough to receive the breath. The whole body is touched by the inhale without effort and the whole body releases the breath without collapse, and nothing particular changes in that process ... knowing this gives you an accurate foundation that will help you refine each and every posture to the smallest detail.
There are a few pranayamas and kriyas that are worth practising separately to explore the choices we have in our breathing, so the body can get used to responding well, but to be honest, if you have the opportunity to laugh, cry, sing, cough and yawn whole heartedly on a regular basis, you have it pretty much covered.
So when I teach, rather than talking about 'inhaling', I would encourage you to find ways to "make space" – then there can be the direct experience of how the breath can flow freely into the posture and the space that you are occupying. I would also talk about how to"gather yourself together to release out" or about "moving out in all directions so that the body collects itself towards the centre" rather than just 'exhaling' – the centred body then has an opportunity to surrender to the support offered by the earth as the breath is released, without leading to collapse, and the spine, free of the pull of gravity, has the rare and glorious opportunity to 'fly'. For me this is Yoga.
(aka Marc Woolford)