In the tradition of yoga which I study in, breath is not forced into any specific region at any time, nor is it constricted. This might vary for certain pranayama, mudras, postures utilizing retention, or bandhas… but as a foundation, the breath should be uninhibited and allowed to be steady, smooth and long throughout all postures. Dynamic movement in asana is done corresponding to the natural movement of breath and spine. For instance, if you are performing an asana which expands the ribcage and opens the chest through extension of the spine, the movement would be done on an inhalation. If you are performing a forward bend which contracts the abdomen and lower part of the rips, this would be done on an exhalation with the aid of subtle contraction of the abdomen inwards to enhance the effect that the movement would already have of pushing the air out of the lungs as the body compresses forward. This also prevents people from moving into postures and trying to round their back and stretch forward against an expanded thoracic cafe. These same principles are followed as one enters into static postures, but in the postures, the breath should be able to move freely within the constraints of the posture (obviously certain postures cause a natural restriction).
In seated or lying breathing, the breath is free to fill the entire lung cavity with ease, resulting in the subtle rise of the chest first and then the slight protrudance of the (relaxed) upper abdominal cavity as the lungs expand outward and the diaphragm descends downwards. There isn’t a force into the chest, like I said - but the focus is for training students who have restrictive breathing habits is on allowing the chest to expand first. On the exhalation, the focus is on drawing the abdomen gently back as the diaphragm relaxes back upwards. It isn’t intense like a deep contraction, but a aid to focus on expelling the air. The chest is relaxed here, with maybe a very, very subtle drawing inwards at the end, so that there is no restriction to the air leaving the chest.
This focus on monitoring the movement of the chest first and then the abdomen (and versa) is often referred to as the chest–>abdomen–>chest, but it’s really just a way to train people to allow the breath to move freely. There is no ‘breathing into the chest, then breathing into the belly’. Once the resistance to full, relaxed breathing is reduced, breathing should be a full, gentle expansion, like blowing up a properly stretched balloon.
When I was first starting in training, I enjoyed laying down and imagining, trying to feel the moment when the diaphragm contracted and began to pull downwards, and then monitoring how the breath would move of it’s own accord. It was sooooo relaxing.
The FIRST school I studied at many years ago, however, emphasized abdominal focused breathing, but I don’t remember that they gave any good reasoning for it. Like Gordon said, every school of thought is different.