Unlike other aspects of Guiding tradition, it is so far unclear what the origin of 'campfire ashes' is. American websites claim a legend that Lord Baden-Powell always took a small amount of ashes from each campfire he attended, and sprinkled them into the next campfire, with the aim of bringing all Scouts and Scouters, from one campfire to the next, into brotherhood. But so far I have found no evidence of this being recorded as happening in the UK. And I would have thought it would be difficult to arrange - ashes can often only be collected the morning after a campfire, when the fire has gone out and the heat died down - often campfires would be part of a busy schedule of events for B-P, he would rarely have the option of staying onsite until the next morning in order to collect ashes.
The custom as it currently stands, is that at the opening ceremony of a campfire, after the fire has been lit, anyone with containers of ashes is permitted to step forward, ceremonially add the contents of their container of ashes into the fire, and tell the gathered group the year of the oldest ashes potentially included, and perhaps also one or two of the key campfires which are represented. The idea is that the spirit of previous happy campfires is brought to this new event. The next day, campers can each collect a share of ashes, which are assumed to contain ash from all of these previous campfires as well as from the latest one. In modern times, a film-canister was used as a handy container, as well as being considered a reasonable share to take.