I have not made #biochar on purpose, but I have one of those Titanium portable rocket stoves, and when I terminate my cooking, I have a small amount of charred wood bits, some charcoal, some biochar and a little ash in various stages... I always dump it into a #10 can and add sand/soil on top to stop the burn - this gets added to my next soil mixing when I make my own posting mix.
As for exporting biomass, even on a microscopic level - it's a given in a "consumers will be mined for profit" environment!