Cardiovascular ability is quite the dirty word in many martial art schools whose worth is mostly in performance in the dojo.
I agree to some extent as investing in the parts of the whole without context gives very little to advance the abilities of the practitioner.
It is putting things together which brings us to a higher level of freedom and ability under PRESSURE.
Runners do not make better fighters necessarily. Doing hard things where your mind and emotions connect with the struggle is what elevates us into calm under pressure.
Breath during running:
Begin running and match your breath rhythm with your steps. Inhale for one step and exhale for one. Increase the count first on the inhale and later on, on the exhale until you reach the apex of the pyramid. Place your awareness in your face, your neck and torso to find out where the tension crawls and shake or release it by shifting your alignment and continuing to breathe.
Begin running and match your breath rhythm with your steps. Increase only the exhale stage of the breath and let it stretch to the max and again find where the tension goes and add listening to your thoughts telling you to inhale more and maintain a steady rhythm for a while until the body and mind find out they can release excess and exist beyond what we believe is our limits.
Begin running and breathe solely through your nose. Inhale and exhale through the nose and focus on maintaining a relaxed face, neck and torso as you nose breathe. Increase the pace of the run as possible (while avoiding moving cars and stray leopards) until you find a new place of silence within.
ADVANCEMENT MUST OCCUR IN ALL OUR LAYERS OR ELSE IT REVEALS THE HUBRIS OF MAN