Breathing and running

In vertebrates, the spine normally lengthens (extends) on inspiration. This extension lifts the top ribs so that the intercostals can lift the lower ribs, and it lifts the central tendon of the diaphragm, so that the action of the diaphragm is primarily in lifting the ribs. In resting breathing, we see a reduction of spinal curves on inhalation, and a slight augmentation of the curves on exhalation. However, this is not what we see during singing, playing a wind instrument, or in running and swimming. In singing, the spine maintains its extension during exhalation so that air is not forced out quickly or with too much pressure. In running and swimming, the needs of the locomotor actions of the limbs require strong, continuous spinal extension.
In general, an increase in the work done by arms and legs will be preceded by spinal extension (as in performing a dead lift, for example). In quadrupeds, one sees spinal extension in the leaping phase of running, and spinal flexion on recovery. Respiratory cycle is linked to gait cycle, such that inspiration occurs on extension only.
This is entirely different in running bipeds, in which spinal rotation is linked to leg extension and recovery. There is no gait extension and flexion, only extension.
Look at swimming, for example. There is a very brief window for inhalation, and exhalation is often spread out through several arm cycles. One needs spinal extension to counter arm pull. Inhalation can be accomplished in less than a second, whereas exhalation needs to be slower to give time for the extraction of oxygen from the air. So, one would expect expiration to be of longer duration that inspiration.
Of course, if a runner is slumped, or too vertical, he is not using spinal extension and will not get the benefit of extension to both breathing and locomotion.
When the spine nicely extends in running, spinal rotation takes care of the transition between leg extension and recovery and back to extension. In the absence of good spinal movement, these transitions are less efficient, requiring some conscious activation. Same for breathing, which, in ideal posture, does itself. In someone with a tight neck and shoulders, for example, the diaphragm will tend to have exaggerated descent, and thus one will see reduced rib movement. Then we see all of these breathing methods appear – ways of breathing better in the absence of good posture. Like all those running methods…

Brief:
In vertebrates, the spine normally (in the absence of injury or poor posture) lengthens (extends) on inspiration. This extension lifts the top ribs so that the intercostals can lift the lower ribs, and it lifts the central tendons  of the diaphragm, so that the action of the diaphragm is primarily lifting the ribs. In resting breathing, we see a reduction of spinal curves on inhalation, and a slight augmentation of the curves on exhalation.
Thus, in healthy subjects with good posture, the downward travel of the diaphragm is limited so that it can effectively lift the ribs, expanding the thoracic cage laterally and in depth. With this lengthening of the trunk on inspiration, we also see an increase in tone in abdominal muscles, which pushes the viscera up against the diaphragm, which helps keep the central dome of the diaphragm lifted to further facilitate its effect on the rib cage.
One certainly does not see, in athletes such as swimmers or runners, a softening of the abdomen on inspiration, as the spine is extended to create locomotion, and abdominals are active in movement, etc. This is well-studied in running quadrupeds, in which the extension of the spine in the leaping phase of running and its extension in inspiration are inextricably linked.