The noun and verb for this kind of behaviour is a stampede. Here's the relevant definition for the noun from the Oxford English Dictionary (not ODO):
2.
a. A sudden or unreasoning rush or flight of persons in a body or mass;
When used in its literal sense, not its figurative one, a stampede tends to imply the injury or death of some people. However, this is not part of the definition. Most people consider the main cause of death during a stampede to be trampling. Here is the relevant definition of trample from OED:
4.
a. trans. To tread heavily and (esp.) injuriously upon; to crush, break down, or destroy by heavy treading; also to trample down , to trample under foot .
However, this is in fact a misconception. Most people who die in a stampede die from compressive asphixiation (see under Forces on page 5 here). In ordinary language this basically means being suffocated by being squashed.
Refs:
"stampede, n." OED Online. Oxford University Press, June 2015. Web. 18 July 2015.
"trample, v." OED Online. Oxford University Press, June 2015. Web. 18 July 2015.