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):
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:
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.
"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.