If you Google up the scientific literature, you'll find that breathing
is specifically influenced, "entrained", by locomotion.  It seems sort
of common sense to think of a racing dog breathing in step with its
strides.  Some people tell us that loads on arm muscles may be different
to those on the legs, although only the same rate of energy expenditure
is inovlved.  This may be relevant for those with weak hearts.

For someone trying to get the best out of their lungs in the sense of
huffing and puffing as little as possible during actual exertion,
something like breathing extremely slowly  by training to reduce the
number of breaths per minute, would seem an option.  Hectic, jittery
breathing would seem less healthy and efficient than relatively slow and
less apparent breathing.  At any rate, such an opinion would be sort of
instinctive.

Doing step-synchronized breathing would therefore have three advantages:

1)    A powerful timing effect (entrainment) and something more than mere
willpower while prolonging breaths when sitting still ,

2)    putting a load on the respiratory system and using it under actual
working conditions and

3)    generally keeping one's breathing under control and avoiding slipping
into hyperventilation likely to herald an attack, for example in the
night.

One Internet poster (Dr. G. W. Cherry of the Massachusetts Institute of
Technology) seems quite enamored with the system for promoting health.
He does not appear to be in it for the money.

See e. g. "My exercise today was breathwalking on Seapoint Beach.
I love this synchonized locomotion and respiration. I dis-
covered the synchronization works best for me when I in-
hale during three steps and exhale during four steps. I also
count the steps subvocally.
Exhale: 1, 2, 3, 4, Inhale: 1, 2, 3, Exhale 1, 2, 3, 4, Inhale: 1, 2,
3...
I become very mindful of the body walking/breathing and
very calm and very happy: Disturbing mentations stop.
My practice is to cultivate this calmness and happiness
in everything I do. Right now, as I think about this calm-
ness and happiness I feel very positive toward you."

I'd guess that a few weeks practice would be necessary.  At any rate it
seems certain that increasing the ability to do 4 plus 4 steps (in and
out) is related to respiratory health.

Prolonging the exhale in relation to the inhale seems to have a very
powerful effect (safety?) for getting mucus in the nose and chest on the
move.  Richard Friedel.