How did they do it? If it was someone important like a king wouldn't they be protectet by a whole bunch of bodyguards during the battle? And how did the other party recognize the one they would kidnap? Was it better to stay incognito on the battlefield, with the chance of getting killed, or was it better to be recognizable as someone from higher nobility but with higher chances of getting kidnapped? And lastley how were they abble to pull a knight from its horse and pull him all the way to their own encampent or was there another way to that?
It's not kidnapping (which implies a form of raptus, seizure and absconding with a person). What you are referring to is the practice of ransom. If a knight, or men-at-arms, was defeated and forced to submit to the person who defeated him he would, theoretically, surrender and become a prisoner in the charge of the person who captured him (the archetypal popular image of this is the offering of a gauntlet as a ritual sign of submission). This submission put the submitter under the protection of the receiver, this meant that the captor was responsible for protecting them and making sure no one killed the prisoner.
The prisoner, post-submission, could then be lead back to the camp and placed under guard, possibly communally although this may just have been a more regular feature of later medieval warfare. Otherwise the person might just stand guard over them. This, as you probably realise, wasn't conducive to disciplined warfare. At Agincourt (1415) Henry V ordered the execution of all of the prisoners taken thus far (which included numerous members of the French high nobility) when rumours spread of a new French army coming to reinforce the battle. This might have been seen as a betrayal of the pledges given by the captors but the French had fought under the Oriflamme (a standard which indicated no quarter would be given) and Henry was legitimately concerned that, should the French reinforce, the prisoners might escape or return to the field.
The capture and ransoming of the nobility was incentivised by both social and economic concerns. The Frankish diaspora meant that, in a battle between different groups of knights, there would be a shared set of sensibilities which would encourage the trust required for ransom to work.
The practice of ransom went, somewhat, hand-in-hand with developments of heraldry. As battlefield insignia became more commonplace and widespread it was easier to recognise who was who. There is very little reason that your average knight would want to go 'incognito' because if they were not recognised for their wealth they might just be killed out-of-hand. It was definitely better to be known rather than unknown on the battlefield, for reasons I explore here. The capture of Jean II, king of France, at the Battle of Poitiers led to a punishing peace treaty (Brétigny, 1360) which involved the ceding of numerous territories, a hefty ransom, and that hostages would be given in exchange for his release (after the payment of a third of the ransom).