In inspiration, as everyone knows, the diaphragm contracts and/or the chest expands. But it isn't as simple as that.
      I want to discuss three nuances: -
I,  as the diaphragm contracts, the abdominal muscles must relax in counterpoint.
II, the external intercostal muscles extend the ribs outwards, while the internal intercostal muscles extend the ribs upwards.
III, with the expansion of the chest the spine tends to extend. So that the eyes can maintain their gaze, the neck compensates by flexing.

I: The Counterpoint of "Abs" and Diaphragm. With contraction of the diaphragm in inspiration the dome of the diaphragm moves down from the level of the 5th rib to the 7th rib.

When the diaphragm contracts, pushing "down" on the organs of the abdomen, the muscles of the abdominal wall (the "abs") must relax to allow the organs to be displaced downwards and outwards. When the diaphragm relaxes during expiration, the abs will contract (back to where they were) pushing the organs back up under the dome of the diaphragm. The abs and diaphragm work in tandem, in counterpoint, reciprocally, rocking the organs back and forth, up and down. (From Kapandji: EO external oblique abdominal muscles. IO, internal oblique, T transversus abdomenus)

II. The rib cage sits under the neck and shoulders. It is limited and tethered, anchored in the neck and shoulders. Because of this, the contraction of the intercostal muscles must tend to raise the ribs and accomplish inspiration. This is equally true for the external and the internal intercostal muscles: they will tend to lift the ribs, and again, as with the action of the diaphragm, the abdominal muscle must comply to allow this movement. However, there is an important nuance and difference between the action of the external and internal intercostals…
     The external intercostal muscles run from the lower border of each superior rib downwards and forwards to the rib below. Because of this oblique orientation of the external intercostal muscles they are going to tend to open the ribs laterally. Meanwhile, the internal intercostal muscles run down and, to a slight extent, to the posterior, but as this angle is very shallow, the horizontal vector of the force will be minimal (though that minimal horizontal vector that there is will tend to oppose the lateral pull of the external intercostals: it will brace the ribs against this lateral expansion to some minimal extent). So the internal intercostals will tend to simply raise the chest wall. It therefore seems apparent that, by and large, the external intercostals expand the ribs laterally, while the internal intercostal raise the rib cage.
     Congruent with this analysis of intercostal action is the fact that the external intercostal muscles are situated more laterally and posteriorly where they can best effect the lateral extension of the ribs, while the internal intercostals are situated more anteriorly and medially where they can maximally raise the ribs (and, therefore, sternum).
     Remember, the rib cage is tethered in the head and neck. Contraction of the intercostals, while it will bring the ribs closer together, does not therefore bring the ribs downward, but rather - the rib cage being "open" at its inferior margin - contraction of the intercostals will (with the compliance of the abdominal muscles) raise the rib cage.
     Note that for the intercostal muscles to lift the rib cage the abdominal muscles must (again, as with diaphragmatic inspiration) comply by relaxing. In inspiration the abdominal muscles relax to comply with and allow the movements of the contracting diaphragm and intercostals. In expiration the "abs" contract to comply with the relaxation of the diaphragm and intercostals. For "active expiration", where more force is required then that provided by the eleasticity of the lungs and gravity on the chest, it will bethe contraction of the muscles of the abdominal wall that will pull the rib cage down.
     There is a second factor involved in the lateral opening and the vertical raising of the ribs, and that concerns the angles at which rib joints are orientated (x and x dash in Kapandji's diagram). In the upper ribs this axis affords more superior/inferior movement: in the lower ribs the axis of the costal-vertebral joints affords more lateral movement. Hence the lower ribs allow for more lateral expansion of the ribs: the upper ribs allow for more superior expansion. (Notwithstanding this regional differentiation in the degree of participation in lateral expansion, it is the external intercostal that accomplishes the lateral expansion.)

III: There are those authorities who say the first movements that we make as embryos are flexion and extensions and, consequently, flexion/extension of the body is the most fundamental of motions. And indeed, where in the adult animal we see extension and/or flexion accomplished with great energy, urgency, and passion, yes, the whole body (back and neck) extend or flex together as a single unit. An example of this singularly integrated flexion/extend is seen in the evasive movements of a boxer (he may flex forward to "cover up"and close himself off, or extend backwards to evade a blow) and again in the extravagant and passionate movements that may be seen in "cathexic release". However, this is not the usual pattern of flexion/extension seen in the body in "normal" movements of breath.
     In the normal movements accompanying breath, inspiration is coupled with extension of the torso (and with external rotation of the upper limb girdle) and, so that the orientation of the gaze of the eyes is maintained, flexion of the neck in counterpoint to the extension of the torso. Likewise, expiration is coupled with flexion of the torso (internal rotation of the upper limb) and extension of the neck - to keep the eyes at a level. And this fact, this arrangement, is recognize in various systems that deal with flexion and extension in the body as, for example, the CranioSacral Rhythms where flexion of the sacrum couples with extension of the occiput, and again in Reichian breathing where we see this same coupling: extension of the pelvis goes with flexion of the head.

I was reflecting upon this and realising that there are several contexts in which the neck and the rest flex and extend together.

The differential functioning of the external and internal intercostals explains practice of "yoga breathing" in which lateral excursion of the lower ribs (external intercostal action) is "isolated" (separate) from the raising of the chest (sternum) (internal intercostals)

Note, of course, that only some of the above is "new". The coupling of abs and diaphragm has been written about as illustrated in the Kapandji illustrations. Likewise the attitudes of the costovertebral joints. The counterpoise between neck and torso, as said, is found in Uppledger's CST and in Reich. What is new (and therefore potentially controversial) is the particular ascription of the differential functions of the external and internal intercostals. The literature either says, simply, that they both function in inspiration, or that the external function in inspiration and the internals in expiration, but this cannot be that case (as argued above).

Note also that there are other muscles involved in the movement of the chest. There are numerous small strap muscles aground the costalvertebral joints, and there is a set of complex "innermost intercostal".