There's unfortunately no first hand account of how such the process worked. There are, of course, Spanish accounts of battles with Aztec and other Mesoamerican forces, but these are hopelessly vague on details.

Hassig, in Aztec Warfare, spends time reconstructing a theory of the order, progression, and resolution of Aztec era battles, which is helpful in understanding the mechanics of how captive-taking may have worked. First, we have to look at the line of battle, which we can best construct as being made up of nuclei of what Hassig tems "shock troops," the heavily armed and armored, experienced soldiers of the military orders. Wielding macuahuitl and shield, they would engage the counterparts and attempt to break the enemy line. Backing them up on either side would be less experienced men armed with tepoztopilli, long thrusting spears.

The line of battle was not static, and reinforcements/relief would regularly filter up to replace the men at the front, as well as specialists who would remove the wounded and killed. This movement provides and opportunity to also transfer captives back behind the battlefront, where they would be bound with ropes. Other methods of securing captives included small cages or wooden collars with projecting sides, which were also placed on slaves with a history of escaping or disobedience.

So, the synthesize all this, the process of taking a captive would typically be initiated by an elite warrior (or plucky junior man) wounding or otherwise incapacitating an opponent. Hassig notes that depictions of captives depict them as young, inexperienced warriors, which he states probably reflected the reality of the situation, with the young men both lacking in combat experience and making up a bulk of the troops. Once defeated, the captive would be seized, with the traditional depiction being grabbed by their topknot, and ferried to the back along with the relieved soldiers, the wounded, and the dead. After the battle, any disputes over credit for the captive would be adjudicated and the prisoner turned over to his captor's care.

All of this is the idealized view of Aztec warfare. In reality, captives would also be taken, and in great number, when the enemy broke ranks and was routed. The number of captives taken in battle could also be conflated with the number of captives take in the sack of a city. For instance, the conquest of Alahuiztlan and Oztoman resulted in a reported 44,200 prisoners, but that's because these cities were sacked, razed, and the entirety of their surviving population taken into slavery, with the towns being repopulated from the core Aztec cities.

Also, despite the widespread focus on Aztec captive-taking, they did fight to kill. It's hard to hold on to that captive you took when you lose the battle and have to flee, and it's not as though you can take a captive with an arrow, slingstone, or atlatl dart. Isaac (1983) "Aztec Warfare: Goals and Battlefield Comportment" describes numerous battles where the Aztec forces wrecked death and dismemberment upon their foes. Obviously, taking a captive was an important goal to an Aztec soldier, and way to achieve advancement and reward, but the Aztecs did not achieve a near century of military success by placing such a goal secondary to winning a battle.