"A pursuit is an offensive operation designed to catch or cut off a hostile force attempting to escape, with the aim of destroying it (...). Pursuit operations begin when an enemy force attempts to conduct retrograde operations. At that point, it becomes most vulnerable to the loss of internal cohesion and complete destruction. A pursuit aggressively executed leaves the enemy trapped, unprepared, and unable to defend, faced with the options of surrendering or complete destruction."
FM 3-90 (U.S. Army), chapter 7
A pursuit usually happens when the opposing forces try to avoid or minimise combat by withdrawal or evasion while friendly forces seek to force a fight on them. Typically, this would be caused by opposing forces feeling unfit for the fight, while friendly forces feel much fitter. It may also be initiated in error, though; the opposing forces may feign weakness and withdrawal, and the pursuing force may run into a trap. The cavalry-centric steppe nomads (Scythians, Huns, Mongols) were famed for such feigned retreats for a thousand, maybe two thousand years.
The typically recommended manoeuvre for a pursuit is not to chase the opposing forces directly (by following the in their trail), but to try to race past them and flank or stop them afterwards. This and pursuit in general puts a very high demand on high speed of movement, while the same applies to those elements of the withdrawing opposing forces that try to provide security for the bulk of opposing forces. Traditionally, (light) cavalry was the branch of choice for pursuit of a beaten army at the tactical level, but at the operational or campaign level entire (mixed) army corps pursue their peers. In theory air and artillery strikes could be used to slow down the withdrawing force, even by scattering anti-vehicle mines. Yet in practice the artillery will likely lack the range and air forces would first need to be convinced to be helpers of the ground forces, enablers of ground forces manoeuvre success, rather than ground forces' rescuers and prime time killers in their own right. To crater a road at a bottleneck would typically mean to not directly drop that bomb on some hostile vehicle for an air force-attributed attrition effect.
"Lediglich frontales Nachstoßen der im fortschreitenden Angriff befindlichen Teile vorderer Linie genügt allein nicht. Der Erfolg wird erst dann erreicht, wenn es gelingt den weichenden Feind- zu überholen- zu überflügeln- sich ihm vorzulegen"
"Handbuch der Taktik, Eike Middeldorf et al, 1957
(A merely frontal push of the attacking elements of the forward line alone does not suffice. Success will only be achieved if one succeeds to
- to overtake
- to outflank
- to place oneself in front of
the evading enemy.)
"It is difficult to intercept and destroy the enemy by merely following the enemy withdrawing from his front and carrying out a pursuit, and it is necessary to push in from the enemy's flank and carry out a pursuit toward the enemy's rear. In other words, it is necessary to be positioned so that the pursuit is carried out from abroad front. In view of the above, in order to intercept and destroy the enemy, pursuit must be carried out over a wide and deep area."
"Principles of War", 1969, p.99
A wide/broad pursuit also reduces the susceptibility of the pursuing force to the feigned retreat tactic somewhat.
Moreover, it's essential to be ready for pursuit (and exploitation in general) at the time of the opportunity. This requires suitable reserves in the right place.
The withdrawing forces are usually not very orderly, not well-secured, demoralised - they are not ready for combat. This offers the prospect of great successes even to small pursuit parties if only those are in a much better state. This is why reserves are so important; those forces that convinced (by combat) the opposing force to withdraw will be morally and physically unfit and would take too much time to regain the necessary order. At least nowadays we could also expect them to be too obsessed about caring for the wounded and some prisoners of war may be burdening them as well.
- - - - -
There is a trivial answer to the challenge of pursuit: Readiness, timing, placing, reserves etc. are all solved problems if only you have a small yet combat-capable force in the back of the opposing forces. Yes, I'm coming back to THIS again. A company-sized element of armoured recce is a sword of Damocles to every opposing forces' manoeuvre commander. It forces 360° security on him, it disrupts his flow of supplies, cuts him off from rotary army aviation support ... and in the event that his brigade runs into trouble and is forced to withdraw said tiny armoured recce element would be ideally placed and highly dangerous for a pursuit action. It wouldn't even need to manoeuvre into position any more. In fact, it would have the choice between maximising attrition by itself and trying to slow down the withdrawing force for a greater pursuit success by larger (and then relatively faster) friendly forces. On top of that, it may cause a panic storm (or accept a disproportionate quantity of prisoners of war surrenders) because the beaten, withdrawing troops would be disheartened by its presence. They couldn't know what caused the noises, the rising smoke columns - and it's quite a stretch to expect forces to withdraw towards or through a threat.
On the other hand, I don't really think that pursuit or withdrawals as described in field manuals should be found in future European-style land warfare. Aside from the fact that there should be peace, evasive manoeuvres and aggressive manoeuvres should be largely indistinguishable until an army formation was really shattered by a well-prepared blow. I suppose that the forces would withdraw in utter disorder (more positive choice of words: "in dispersion"), which is the quickest method of movement and the one that could be stopped the least. This assumes that no obstacle such as a river provides bottlenecks through which only organised forces could pass safely (through engineer efforts or by fighting through an occupied bridge). A somewhat orderly withdrawal of large forces smells very much of the foot-mobile armies of the past. Fully motorised forces would flee very differently.
Coming back to the 'reserves are important' thing: A battalion battlegroup or brigade appears to be the army formation of choice for conventional land war, and usually it would split its combat vehicles into no more than two elements (afaik). There's very little reason to believe that the commanding officer of such a manoeuvre element would keep reserves for pursuit - any reserves would likely be employed to achieve the effect that convinces the opposing force to withdraw in the first place.
"Reserves" don't need to be organic, though. Another battlegroup nearby (less than 50 km away) could be tasked with the pursuit by a higher CO (division or corps level). What matters is that suitable forces are available for the job. It's old-style division-centric thinking to expect these reserves to be held ready by the same CO who led the effort to compel the opposing force to withdraw.
So once again, having multiple manoeuvre elements puts an advancing opposing force at great risk: Two elements could defeat it and force it into a withdrawal with a coordinated two-axis assault, while a third could mount a devastating pursuit. This prospect should serve as a great deterrent, since advancing with one or few manoeuvre elements would put them at great risk. This is somewhat reminiscent of the function of a front line: It would take a concentration of forces (that's not easily overburdened by 360° security and not easily thrown into disorder and ruined by a pursuit) to advance. A couple of manoeuvre elements could be threatening enough to deter an advance by opposing forces this way, without actually getting into contact.
You'd need powerful manoeuvre elements for the assault, but weaker/smaller ones could do the pursuit.