It is true that breathing is an integral part of singing, but it is not the main aspect of sound production; air simply gets the vocal cords to vibrate. Think of breath support as a pipe organ; the wind pressure remains constant and the sound is produced when a key is pressed and the air gets a pipe to “speak”. The blower does not start and stop for every note played. Singing running notes from the diaphragm equates to the blower starting and stopping for every note, and only serves to make them heavy and unreliable, not to mention using the breathing muscles in a way they are not meant to be used. Instead, take a breath, starting at the abdomen, then filling up the lungs (but never raising the shoulders) and let the air out at a pace according to the vocal needs, keeping a constant airflow. Note attacks, releases, and fast running notes all happen in the throat, not in the diaphragm. For precise, accurate, and efficient running notes, keep the air flow constant and articulate each note from the vocal cords. To develop this agility, work on runs slowly, and over time speed up the process until you can sing each note clearly and easily.
Building breathing capacity and breath control requires an integrated approach. The idea of separating breath from voice makes no sense. Athletes do not work on their breathing in preparation for a marathon. They increase their physical strength, lung capacity, and breath control by running a little longer distance every day.
From our knowledge of muscle building, it has been demonstrated that short, quick movements provide little change compared to slow and controlled movements. The same goes for the voice; to develop vocal strength, breathing capacity, and breath control, use long tones, increase the length of held notes over time, sing slow scales (preferably descending), and sing longer phrases during vocal training. The result is that, as the voice is developed, less air is needed to produce a healthy sound.
Stop wasting time on useless breathing exercises and focus instead on producing a nice, healthy tone. Breathing is really easy: breathe in, breathe out. It really is that simple.