I want to know how far back a modern American peer or near peer military infantryman would have to go to completely change the tide of a specific battle.
I don't think the modern military infantryman is that much more well equipped than those from Vietnam or Korea to turn a loss into a win, or even as far back as WWI. Machine guns were already a thing, grenades were common, and I don't think an M4A1 carbine is that much better than WWI weapons for one man to change the tide of the war. However, if we go back to the Civil War we are getting to the point that an M4A1 carbine simply outranges and is more accurate than Kentucky rifles that now you can pick off officers from out of range of anything else, and that could do it. If we go back all the way to the Revolutionary war, one man with 2000 rounds and an M4A1 carbine is absolutely shredding the old rank and file musket lines, and I think they would likely be able to turn almost any Continental Army loss into an American win.
I'm going to use American terms as I am American and that's what I know, but I don't want answers limited to America.
What I mean by "average infantryman": an average Joe in their branch of military. In America, I'd say someone below an E4 who is not on track to join a special forces branch and is 2-3 years into their first contract. This person could be from any branch that has infantry, so army or marines.
What "with their kit" means: Whatever is standard for someone in that branch to carry, including primary and secondary firearm (so M4A1 carbine and sig P320 [though I believe that is under investigation, so let's say the baretta that's been in use for the last while instead] for the US army), grenades, armor, medkit, nods, etc. I will add the caveat that their firearms get a couple thousand rounds, not the couple of magazines they currently carry.
Other caveats:
They can only go back as far as their branch, or a direct predecessor in their area, exists. US soldiers can go back to the British army in the French Indian war if that's as far back as you'd need to go. However, the Continental Marines were formed in a bar in 1775 with no direct military predecessor, so no US marines in the French Indian war. France can use the French foreign legion for as long as that has existed, but can take a soldier in the French army as far back as there was a country whose name translates to france that had an army that could be a direct predecessor of the modern French army. My understanding is that the British Army came from the English army, which has existed for over 1000 years, so all of that is fair play. However, there is no direct line, afaik, from the modern Italian military to rome, so that's out.
By completely change the tide of the battle I mean make the losing team win, or turn a draw into a win. An American soldier would have to make the US Continental army with the battle of Bunker Hill, or a French soldier would have to make France win the battle of Agincort.
I'd also like to keep it to specific, well known, military engagements. No saying a Tennessee state trooper would change the outcome of the battle of athens, as that is neither well known nor military. Think something a high school teacher would be likely to mention. The Falkland war is hardly mentioned in America, but I'd bet is in Britain, so a specific battle from the Falkland War is game. Almost no Indian or Chinese wars are taught in American high school, but whatever is common knowledge there is fair game as well.