What kills is cessation of brain activity. That can happen two ways. The most obvious way is destruction of brain and spinal cord by a direct hit. If the brain is destroyed, death is immediate. If the spine is destroyed, everything "downstream" of the hit will stop working. If what's downstream includes the heart and lungs (like a neck shot) death will be quick. They will stop working and the brain will starve for oxygen. If the hit is farther back, a finishing shot will likely be needed.
If the heart lungs are destroyed, blood pressure goes to zero, and the brain starves for oxygen in a few seconds. During those seconds, an animal as fast as a deer can travel a long way. Sometimes it does; sometimes it doesn't. But it is not "shock" that kills anything.
Sometimes, depending on the state of the animal and the specifics of the particular shot placement, the animal may drop at the shot. It seems to me that the level of hydrodynamic shock delivered does have some correlation to whether or not this happens, but I don't think it is clearly understood what exactly is the cause, or what causes failures to drop the animal instantly. There are theories, of course, but I'm not going to try to describe or defend any.
I do want to make it clear, however, that, although hydrodynamic shock is part of the effects a bullet produces, it is not a wise idea to think about it as a primary part of what kills an animal. What kills is disruption of brain function. Until that happens, the animal will be alive. Destroying the heart and lungs works so well because it will always stop the brain in a few seconds, and sometimes it will even include the sudden DRT bang/flop. But it is more important to think about how to disrupt blood flow to the brain when you are shooting an animal than it is to hope you have a certain level of kinetic energy.