My unscientific opinion on what kills game is thus:
1. Damage to the central nervous system. Either with damage to the brain, damage to the circulatory system that leads to asphyxiation due to lack of oxygen rich blood reaching the brain, or damage to the respiratory system that leads to asphyxiation due to interruption of the gas exchange that occurs in the lungs leading to a lack of oxygen rich blood reaching the brain, and damage to the portion of the spinal cord that sends signal to the circulatory or respiratory systems.
2. Damage to other systems such as the renal, excretory, digestive, or urinary which can lead to malnutrition, sepsis, immunodeficiency, etc. This can be caused by direct tissue damage or as a result of circulatory system or nervous system damage.
3. Infection and disease.
The speed at which these things result in death or incapacitation depend on any number of things including what other trauma was incurred during the generation of the wound, the size of the game, the resilience of the game, the size of the wound, which system incurred the damage, etc.
It's interesting how game reacts to being wounded in different ways. Everything from instant incapacitation to a slow painful death to full recovery. I've personally never had an animal fall where they stood from a lung/heart shot, only from a central nervous system shot, regardless as to what type of bullet I used. Terminal performance is not always textbook.