Tennessee v. Garner is the relevant Supreme Court ruling here. An excerpt from the holding:
The Tennessee statute is unconstitutional insofar as it authorizes the use of deadly force against, as in this case, an apparently unarmed, nondangerous fleeing suspect; such force may not be used unless necessary to prevent the escape and the officer has probable cause to believe that the suspect poses a significant threat of death or serious physical injury to the officer or others. Pp. 7-22. [471 U.S. 1, 2]
Bold and red by me. IANAL, but WaPo's Volokh Conspiracy blog (which is mostly lawyers and law-related stuff), pointed me to this.
This is why I think the prosecution has an uphill climb if certain facts turn out to be true. If in fact, DW hears over the radio about the strong arm robbery and realizes that the guy he just rolled past fits the subject description and actually appears to be holding the stolen merchandise, he has reason to be wary about confronting them. OTOH, as an officer for six years it's probably not remotely the first time he's had to use caution in approaching a suspect. I don't think if just at that point Brown had cut and run you could argue dangerous fleeing felon. Or you could, but the argument would be questionable. However if the officer was assaulted, if there was a struggle for his gun, I think without question you've now met that criteria.
Which means that even if the shot in the arm came from the back as some say, legally, it could well be justified. The only way it could be considered unjustified is if Brown had quit running and surrendered ("hands up"), and was then shot, but the witness accounts don't seem to support that. A criminal conviction means they must find that beyond reasonable doubt Brown was not attempting to flee and that beyond reasonable doubt the officer was not in reasonable fear of his own immediate safety or the safety of others in the community. I don't see them meeting that burden right now. But that's all just me, and like I said, I'm not a lawyer, so take it for what it's worth.