I was told they were coke fires by a very old bearded gentleman tending to one when I passed once and asked. As for the logistics of transporting it there, I have no idea.
He was perhaps "having you on"
Coke production is more industrial (and isn't made like that), nor a cottage industry like you saw there.
BTW, have you seen it more than once?, I would suspect it's only a very occasional process?.
Like you say, why bother transporting it there (not exactly an easy place to drive a truck to) when B&Q sell charcoal in bags - mass produced somewhere but I doubt in the Derbyshire Dales.
It has to come from somewhere
Perhaps you weren't aware?, but before the advent of coal in iron making, charcoal was used instead (the invention of using coal instead was made in Ironbridge, Shropshire), and this devastated huge areas of forestland in the UK where charcol was produced, just as you saw in Lathkill Dale.
Maybe there are natural coal seams in Derbyshire. I have no idea of that either. In Lathkill Dale, there are certainly many deep man made tunnels into the rock faces. Just past the crossing in the Dale from the steep road upto Over Haddon.
No coal mines
You were lucky to grow up in such a nice part on the country. I grew up on the edge of the West Pennine Moors in Lancashire.
Aren't the Moors nice there?, I grew up on the edge of Stanton Moor, at a small village called Birchover.