These two are intimately related, so can’t mention one without the other.

To be able to run or walk we need to breathe, and to recover from exertion we need to know how to control our breathing.
We should learn to breathe and move, not move then breathe.

Good posture helps us to breathe more efficiently.

This is where we are stacked: head on top of chest, chest on top of pelvis, pelvis on top of feet.
We need to be balanced so our weight can be centred.
A certain amount of joint movement, strength and stability is needed to run. This is mostly in the hips, pelvis, knees and ankles, but trunk mobility is also important. Restrictions in mobility can prevent us from running in our best posture possible, restrict our breathing and contribute to injuries.

Good Running posture:
1. Think tall, so the head is over the chest, and chest not stuck out or slouching forward. This reduces pressure to the lower back.
2.Weight best in the middle, not leaning back through heels, but not running on toes. This helps body weight to be over landing foot and prevents over striding. 
3.Lean slightly forward from the ankles, not from the hips or chest.
4.Hands and shoulders relaxed, not letting hands swing across the midline. Left and right hand movement should be even.
NB If the arm swing is uneven and rotates one way, then the pelvis will rotate in the opposite direction. This can lead to unbalanced load through the pelvis and whole lower limb, leading to any number of issues. This is something easy to think about while running.

The ideal automatic breathing pattern at rest is very slow, light, and mainly abdominal (diaphragmatic breathing). 
The diaphragm is the principal muscle of your breathing. Poor posture can affect the movement of the diaphragm restricting its height and depth, thus reducing oxygen intake. The lower parts of the lungs are about 6-7 times more effective in oxygen transport. When we breathe faster we breathe more air, we get less oxygen in body cells. Thus, the slower your breathing, the more oxygen delivered to where it is needed.
To get a deep breath, we breathe in, the diaphragm moves down as the lower ribs, swing upward and outward like a “bucket handle” increasing the size of the chest space from side to side. As we breathe out, the actions are reversed, the diaphragm moves back up and the chest space reduces.

Although running doesn’t allow continuous deep breathing, proper breathing will help keep the breath balanced. In running we mostly breathe in and out through the mouth, but if we think about how the ribs and diaphragm move, and using the abdominals we can focus on keeping our breathing slower and more even. Any breathing through the nose slows out breathing, forces air down lower and gets more oxygen to where we need it most, our legs! This will make it easier to maintain a nice rhythm and more able to adjust to faster bursts.

Then during your walk or recovery periods think about your breathing. Breathe in through the nose, push the lower ribs out and up, then slowly exhale (mouth or nose) getting air down to the bottom of the lungs.
This will slow your breathing, recovery will be easier and the heavy gasping should be a lot less!