From Coherent Breathing - The Definitive Method, a description of the thoracic pump.
The "thoracic pump" is the name given to the functions within the green box. It comprises the thoracic cavity, the diaphragm, the lungs, and the heart.
The diaphragm governs the pressure inside the sealed thoracic cavity. As it moves down, pressure in the cavity decreases and venous blood rushes through the vena cava via the right heart into the lungs. Pulmonary blood vessels expand dramatically, filling with blood, air and blood meeting across the very thin alveolar surface. The deeper the inhalation, the more negative the pressure, the more blood flows, and the fuller the lungs become.
As the diaphragm moves up the pressure in the thoracic cavity reverses. Pulmonary blood vessels shrink ejecting an equal volume of blood out of the pulmonary veins into the left heart. The left heart raises the pressure and checks and regulates the flow. The more complete the exhalation, the more positive the pressure becomes and the more blood is ejected from the lungs.
When perfected, the "Valsalva Wave" (arterial/venous wave) rises and falls with exquisite regularity and coherence. The left is a view of the Valsalva Wave in the capillary circulation, in this case the ear lobe, where we see the wave rising during exhalation and falling during inhalation. The faster wave riding on the slower wave is the heart beat. At the right, we see the blood volume in the medial cubital vein falling during inhalation and rising again during exhalation. The falling is principally a function of decreasing pressure in the thoracic cavity during inhalation. Notice that in the venous view we see no "heart beats", i.e. would-be pulses of the right heart.
Valsalva Wave as viewed at the ear lobe Valsalva Wave as viewed at the medial cubital vein
When inhalation and exhalation are equal, equal volumes of blood fill and empty from the lungs with each cycle of respiration. Autonomic functions work to insure equality in blood entering and leaving the lungs. If there is inequality, blood may accumulate in the lungs resulting in pulmonary edema.
The heart functions as a bidirectional pump, regulator, and check valve.
As the rate of inhalation and exhalation increase, pressure excursions within the thoracic cavity are reduced and the volume of blood flowing into and out of the lungs as a consequence of changing thoracic pressure diminishes. As this happens, the burden of circulation is placed fully on the heart and the arterial system, this includes venous return, which is otherwise dominated by the thoracic pumping action vs. the action of the right heart.
Conversely, as breathing slows and deepens, thoracic pumping action re-assumes a significant role in generating circulatory force, thereby relieving the heart.
From this we can say that when breathing is very rapid and shallow, the heart bears a very large part of the burden of circulation including venous return. When we breathe slowly and deeply, the thoracic pump offloads the heart - especially the job of ushering blood through the low pressure venous system.
In summary, there are 2 (really 3) motive forces of blood in the circulation:
1) The thoracic pump, which contributes strongly to circulation when breathing is slow, deep, and rhythmic - especially venous return,
2) The heart, having the primary role of elevating arterial pressure and checking and regulating arterial flow, and
3) The muscular action of the arterial tree which relaxes during exhalation to accommodate a large volume of blood exiting the left heart upon exhalation, and constricts during inhalation moving blood along the arterial tree as blood flow into the aorta is diminished during inhalation.