My question is for any modern war (ie: with gunpowder, rifles, etc.), but I'll use the battle of Smolensk in 1941 to further illustrate my question.
I've read / heard that the Germans captured 300,000 Soviet soldiers. At what point do the Soviets know that they've been defeated? Does their leader just decide at some point to communicate to his men to all surrender and they just throw up their hands and hope not to be killed?
Now that the Germans have captured 300,000 men, what do they do with them? That would take a lot of logistics to bring them all to a prison of sorts. Do the soldiers who won the battle take care of this?
I can speak with regards to current military. There are Standard Operating Procedures with regards to POWs, and in any attack, from 8-man section to a 10,000-man division, the first thing done is called an Ammo/Cas report. That stands for Ammunition and Casualties. This is done by a specific member of each formation, and the information is passed up to higher.
You find out how much ammo was expended in the attack, the current condition of your soliders (injured or killed), any POWs, condition of your equipment and so on. This information is collected by the section or squad, and passed up to the platoon, then company, then battalion, and so on. There will also be designated CCPs or Casualty Collection Points. Casualties are passed back the lines (or stay in place as the assault moves forward) and are handled by medics, then evacuated to hospitals.
POWs are sent to a collection area for POWs, and are disarmed and searched, then passed on to a company level, then battalion, and onwards to Military Police.
The Geneva Convention on the Treatment of Prisoners of War is very specific about what you can and can't legally take from a POW (weapons and ammo: yes, gas mask: no), about treatment, handling, pay (yes, you must pay a POW in your custody) and so on, all of which must be followed.
Once in the custody of Military Police, they are taken to designated facilities with are usually run by MPs, who also receive more extensive training on POW handling.
So, yes, soldiers on the battlefield will handle this. It's usually the second in command of a group who is responsible for Ammo/Cas, so the Section 2IC, then the Platoon 2IC, Coy 2IC and so on, and they get collected, passed up, collected, passed up, and so on until they reach a permanent holding area. There are tons of operating procedures that deal specifically with this, and soldiers at the lowest level will know the basic of it. On military leadership courses you get more training about what specifically to do, and you will get more training in your Rules of Engagement (ROEs) about the specifics for that particular theatre of operations before you deploy.
Examples of Ammo/Cas reports: https://rdl.train.army.mil/catalog-ws/view/100.ATSC/04183AF4-34EB-47F0-BCEE-29C93432DA49-1274564010088/3-21.8/chap6.htm
SOPs for Casualties: http://www.armystudyguide.com/content/army_board_study_guide_topics/survival/infantry-platoon-tactical.shtml
Geneva Convention on POWs: https://www.icrc.org/ihl/INTRO/305?OpenDocument