As I understand, in Europe, after an enemy force was defeated, post-battle pursuit would be conducted by heavy cavalry armed with sabers and lances. The armies in the American Civil War, to my understanding, lacked a comparable heavy cavalry force, with most of their horsemen being mounted rifles. Would these mounted rifles have conducted the pursuit, or would it be conducted in a different manner, if at all (looking at you, McClellan)?
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One of the aspects of the American Civil War that has been stated pretty forcefully in most of the historical literature is that, while both sides sought to destroy or capture each other's armies, the armies themselves were, in practical terms, indestructible. Even after monumental disasters like Fredericksburg, or intense maulings at Shiloh or Antietam, armies proved resilient, cohesive, and if not combat-effective, at least capable of maintaining a fortified camp. Both sides, at various times, intended to -or attempted to - pursue enemy armies, both in a tactical and strategic scale, and both sides used a variety of force compositions and tactics to do so. But "pursuit" was often much more of a theoretical thing than it was a practical one, and I think someone much more fiery than McClellan would still have had a difficult time pursuing the rebels after Antietam, for a variety of complicated reasons that have much more to do with the nature of battle in the Civil War than it did about force composition.
I'm going to talk a bit about tactical pursuits first, in which smaller, more localized engagements involved pursuit of fleeing enemies and the capture or recapture of tactically important terrain features or fortifications, and then talk about some of the larger-scale pursuits, or attempted pursuits, to show how strategic pursuit was intensely difficult in the Civil War. To keep things relatively limited, I'm going to discuss this mostly in relation to the Battle of Gettysburg, but there are a great many examples throughout the war we could get into in follow-ups.
Tactical Pursuit
"Pursuit" was a term used very often by officers on both sides during the war. It was a central aspect of warfare at the time, and was a means by which information, prisoners, or localized advantages could be seized or secured. When an enemy force was in retreat, they were vulnerable. If your force had the means to pursue, you did so with as little hesitation as practicable. Pursuit meant taking a tactical advantage, asserting a tactical threat. The enemy has to turn and stand, because the alternative was to be chopped up from behind. If they had an obstacle in their path - a river or mountain or suchlike - you could trap them, surround them, and demand their surrender.
Surrender was a key feature of the American Civil War. According to David Silkenat, "the number of soldiers who surrendered during the Civil War is approximately equal to the number of soldiers killed." As many as one quarter of all men engaged in the Civil War surrendered at some point. Prisoner taking, the granting or refusal of parole, and the expectation of fair treatment by surrendered troops were all relatively formalized during the war, and surrender was so common that dozens or hundreds of men could be taken on the field all while battles waged for hours or days after their taking. If tactical pursuit had any goals over and above securing tactical advantage, taking prisoners was certainly among them.
The nature of these little pursuits was pretty simple. At some point during an engagement, one side or the other decided that it had had enough, couldn't hold their position or couldn't take the enemy's, and decided to withdraw. If the enemy was strong enough, fresh enough, or eager enough, and had officers who recognized the enemy's withdrawal, they would begin their own advance. While we might have visions of ferocious charges with fixed bayonets or saber-waving horsemen, the reality was often an orderly, organized advance. The idea at this stage was just to put pressure on the enemy, and force them to stand and fight in a weakened state, or compel them into a more disorderly retreat. This could be dangerous for the pursuers, because if you misjudged the enemy's state of cohesion, you could easily just be leading your men into a slaughter as they left a defensive position and marched out into the open. As such, the decision to pursue, or not, was a tough one, and experienced officers as often as not erred on the side of caution.
JEB Stuart, not known for his caution, made the decision to hold off on the pursuit of federal cavalry withdrawing across the Potomac after the skirmish as Brandy Station in the prelude to Gettysburg, because the union cavalry was supported by federal infantry, and made pursuit a potentially costly decision. In another example, though, during the collapse of Pickett's Charge just days later, Pickett's straggling elements were swept up by "clouds of blue-coated skirmishers" who took dozens of prisoners in their counter-charge.
Examples like this are all over the Battle of Gettysburg, and gives some insight into the nature and scale of these battles. While we call it the battle of Gettysburg, the conflict itself was an interlocked, overlapping series of localized fights for terrain strongpoints. Every portion of the battlefield had ebbs and flows, charges and counter-charges, captures and recaptures, and with it the requisite tactical pursuits.
Even when your side had the advantage - it was the one doing the capturing, the chasing - this was hard work. A pursuit could be a leisurely stroll of armed, organized men into the midst of shattered, shell-shocked men and simply pointing them to the rear, or it could devolve into a chaos of individual fights with individual men. A pursuing force could easily lose its own cohesion, unless it was carefully judged and commanded, and a single-minded pursuit could be taken unawares by an unexpected counter-charge, as its own cohesion and organization could be easily diffused by over-excited men running off to their own ends.
I'll describe one example at length, just to put this in better perspective. On the second day of battle, a rebel charge into the Wheatfield pressured the federal defenders to the extent that some whole regiments were ordered to withdraw without challenge. A federal sergeant in the regulars later wrote of the disbelief he felt when, instead of the expected order to charge, he was ordered to about face and withdraw, and he and his men were "cut to pieces." A nearby regiment, also a regular, reported that they'd lost half of their men to shots in the back and captures without inflicting any damage on the enemy.
The regulars withdrew, if we're to believe them, in good order, contrary to the pell-mell chaos and disorder of the volunteers, while behind them the rebels pursued closely. But their pursuit and sense of overwhelming victory had damaged their own cohesion, and as federal troops rushed to their rear, a line of federal guns, the 1st Ohio Light, were charged with double canister and under steady leadership. Artillerists stood tall and waved their hats at the retreating friendlies, shouting at them to get down and take cover. The battery fired through the scatter of retreating federals and into the pursuing Georgians, checking their charge. Several federal survivors described their narrow escape, and one wonders how many weren't so lucky. Grim though it was, the 1st Ohio Light and other guns and stable troops did check the retreat, and prevented a general rout from the Wheatfield, but afterward, neither side was in much shape to charge or pursue. They were exhausted, bloodied, and disorganized, and needed time to reconstitute into a fighting force. Descriptions of men too tired to attempt a pursuit litter the literature of the Civil War, and mark nearly every battle, if not every engagement, of the war.
Strategic Pursuit
This kind of engagement was a microcosm of the kinds of problems that whole regiments, brigades, and divisions had in attempting the pursuit of like segments of enemy armies. At this scale, pursuit could only be practicable with segments of the army which had been unengaged, or only very lightly engaged, during the earlier battle. Understrength and exhausted men would quickly tire out, and if they succeeded in catching up to the enemy would likely not present much of a threat in any case. If an army had reserves, and the enemy itself was in flight, terrain, logistics, and potential cost were all issues that had to be worked out.
After Gettysburg, there was a serious discussion about how to follow up. That Lee's army was in retreat was clear, and that they had taken a brutal mauling was clear, but Meade's army was hardly in great shape, either. Both sides had committed nearly their entire force to the battle, and there were very few fresh forces for either side to throw into a pursuit or a rearguard action. Nevertheless, the Union high command considered a variety of approaches to try to turn this victory into something more meaningful. One plan considered sending a force to the crossing at Harper's Ferry to challenge the rebel crossing, which would pinch Lee between a blocking force and the main body of the Army of the Potomac. The plan was eventually scrapped, but it was considered.
As it was, the exhausted men of the federal army were pulled onto their feet and kept marching in the footsteps of the rebels. Meade reported to Lincoln at one point, four days after the battle, that the Eleventh Corps had marched 30 miles overnight and during the day. A soldier in the Iron Brigade wrote that "the hurried pursuit of the enemy to this point has been by far the most trying campaign.... The men have become ragged and shoeless, thousands have marched for days barefooted over the flinty turnpikes." Porter Alexander, on the other side of the pursuit, wrote that "nothing was more wearying," having to stop every few miles to clear an obstacle or to re-organize a force or to take a short break, or not.