I recall learning in grade school that both the British and American armies would station a line of troops behind the main line, with orders to kill any soldier that turned around and ran during a battle. I was thinking about this last night, and I can't imagine how this would do any real good for an army. I would imagine that this would be disastrous for group morale and trust, to have one portion of the army assigned to kill their own comrades. I can't imagine this being good for any sort of social organization. Moreover, were the armies so poorly trained that soldiers were liable to flee en masse? How did you convince one segment of your army to kill their buddies?
So my question is, if this is true, then how did it really go down in practice? And if it's false, why was I taught it?
The rear guard's main job would have been mostly to act as a reserve rather than to execute deserters. Knowing that there is a reserve would have actually been good for morale.
In terms of what was done with deserters, as The1Man notes having someone stand behind the line and in some way discipline those in front wasn't that uncommon. In formations like those the sergeants often had the job of pushing troops into place to fill gaps that might open due to casualities for example.
But it wasn't simply a question of killing those that ran. Most of the time it was just a way of stopping those that flee from running until they could run no more - instead they would be stopped short. At that point they could be turned around, or if it were a large enough rout, the troops could be made to regroup. MPs can of course arrest deserters rather than shoot them. The threat of force was there, but that was more an insurance thing than the default solutionn to all problems.
John Keegan in his book The Face of Battle suggested that flight from modern battlefields might be less common because there isn't anywhere to flee too - armies are pretty good atd collecting people trying to escape the battlefield (also the range of weapons make it a lot harder to get away from the enemy, but thats another story). His conclusion is probably a bit of a stretch, but he has a point.
I can't speak to the Revolutionary War specifically for some of these questions, but this practice or practices similar to it are common throughout history. Officers have traditionally carried pistols as sidearms despite the fact that rifles have better range and power. That's because they don't need to be able to shoot very far - the pistol was for executing deserters, who tended to be standing right in front of you. Spartan soldiers were traditionally sent off by their mothers who would present them with their shield and instruct them (essentially) "With it or on it". Because one of the first pieces of equipment a deserting soldier would abandon was the enormously cumbersome shield, the intended message is that they should return either victorious, or dead (being borne on their shield). Returning without the shield would only bring ostracization. In the Cimbrian War the Roman Republic faced Germanic and/or Celtic tribes who, when giving battle, would have their women spectating and armed. Their role was to encourage the men during the battle and, in the event of a route, slaughter as many of them (and the pursuing Romans) as possible. It was an all or nothing strategy that encouraged soldiers to hold back nothing - if they didn't win, they lost - everything. No round 2 after that.
This has essentially been the logic of this sort of tactic from the get go. Why do soldiers run in battle? They're afraid of dying and they think running gives them a better shot at surviving. So let's change those odds around and put a line behind them specifically to kill deserters. Well, now if you fight you might win and live, but if you desert then suddenly your enemy is on both sides. If survival is the endgame, your best shot is suddenly to give it your all in overcoming your enemy, because the choice is no longer "maybe die fighting or probably not die retreating," it's "maybe die fighting, but almost definitely die trying to retreat". The goal is to make the consequences of retreating even more dire than the consequences of seeing the battle through. This mentality could mean the difference between a few guys slipping away and a full on rout.
To address some of your other questions, and speaking of routing (consult www.en.wikipedia.org/wiki/Rout because I'm having trouble doing a proper hyperlink for some reason), yes, troops were very liable to flee en masse, and training didn't always have that much do with it (though do remember that in the case of the Revolutionaries, a lot of these men are not soldiers by trade, which exacerbates the chances of a rout). You start seeing men running away, that hurts morale. Why's that guy running? Is the battle going badly? Oh man, it's probably going badly. Should I be running? I don't want to die. How much am I supposed to give for this war? I'm just a guy. I'm running. To quote the wiki article I linked:
A routed army often degenerates into a sense of "every man for himself" as the surviving combatants attempt to flee to safety. A disorganized rout often results in much higher casualties for the retreating force than an orderly withdrawal. On many occasions, more soldiers are killed in the rout than in the actual battle. Normally, though not always, routs either effectively end a battle, or provide the decisive victory the winner needs to gain the momentum with which to end a battle (or even campaign) in their favor.
Preventing a rout can be the difference between victory and defeat on a huge scale. So yes, it's probably not great for morale to have men assigned to kill deserters, but if the option is to take a blow to morale in exchange for reducing the possibility of a rout, a lot of armies would, and did, take that choice. As for convincing one segment of the army to kill their buddies, well - who says they're buddies? Armies are big. Under this sort of military philosophy, a deserter stops being on your side the second he turns to run. They're just shooting down another enemy soldier. The threat of being shot down would (ideally) convince enough men to fight on instead of retreating that routs could be avoided.
Again, I don't know if this happened with the Revolutionaries or how it might have played out, but it's not unrealistic and certainly has historical precedent (and even postcedent).
Edits because omissions and formatting.