What I mean by champion dueling is the leaders or champions of two armies facing up in front of their respective armies before battle and well, dueling. I see this pop up a lot in accounts of early Islamic history, with Khalid bin Waleed and Ali Ibn Abi Talib being particularly prominent examples. The circumstances of these events lead to me to take these accounts with a grain of salt. But I've also seen an example of a certain Roman general, Marcus Claudius Marcellus, who killed a Gallic chieftain in single combat. However, it seems that even contemporary Romans considered this an extremely rare and exceptional event. So was it very uncommon? Did it happen primarily in more tribal societies, such as those of the Gauls and of the Arabs? Is that why it's somewhat rare amongst more settled societies? If it did happen, what was it's purpose? Merely to raise morale? Or was it so common that if a leader didn't do it, they'd suffer a loss of face? Sorry for the long question.
These kinds of duels seem to have been a feature of Greek warfare in the early Archaic period; I wrote more about this here. The idea was usually that the duel should serve as a substitute for all-out battle. Since wars were often caused by personal conflict or slighted honour among the elite, and since social status was largely determined by martial prowess as well as wealth, it made sense for the leaders of armies to make a show of personally settling the dispute. This is also why, both in the Iliad and in epic stories from other societies around the world, battle is often described as a vague and irregular clash in which the important bits are the match-ups between named heroes. However, since the practice of duelling to decide battles was open to abuse, and since the result was always disputed, Greek communities abandoned it in favour of more organised mass warfare.