Speaking only for US Cavalry tactics, one out of every four men remained with the horses to control them.
Pistols and sabers were the perferred mounted weapons, as at least one hand would be necessary to control the reins, this was also one of the reasons that cavalry men had that thick leather strap around their shoulder that was attached to the carbine, you could fire one round and then drop the carbine (leaving it dangling) and draw pistol or saber as necessary.
In modern cavalry, the troopers would ride into the area, dismount, have the horseholders lead the mounts back out of the line of fire, the dismounts would deploy as skirmishers and start fighting. If the enemy started to withdraw, the horses were brought forward, and then used to bound forwards to the next position.
With the limitations on weight that a horse can carry, support weapons would either be carried on pack horses or in wagons. Your T2000 cavalryman would have a pistol and either a SMG or an assault rifle with maybe a M203 as the largest weapon carried.
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The reason that the American Army does so well in wartime, is that war is chaos, and the American Army practices chaos on a daily basis.
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