I did this the other way around whilst walking the Border from Carlisle to Berwick a few years ago. I camped at the campsite at Kielder and, having some foot pain, decided that the bridleway from Kielder to Byrness would be easier than walking on the ridge on the next day. This turned out to be wrong (!) The bridleway gets rougher and rougher and would be impossible for anybody on a horse or a bike. There's a river crossing and deep bracken on steep slopes and I was a bit grumpy by the time I turned up at the hostel (as it was) at Byrness. They soon cheered me up, however and I continued towards Berwick. A year or so later, on halloween as it happened, I filled in the gap on the Border ridge that I'd missed and found the going boggy but much easier. I camped near Carlin Tooth and continued past Carter Bar, descending to Byrness from the North-East on a forestry road that handrails a bridleway which is impossible to follow due to forestry. I returned to Kielder the next day on the toll road, (to be met by a police lady who told me that somebody had reported my car abandoned and that I'd gone into the forest to top myself (!))
Anyway, the point is that following the Border fence along the ridge is much easier than the bridleway and much more fun. I would advise, though, not to descend to the Kielder Stone as its very rough around there. Keep to the ridge to Deadwater Fell. There's an old railway line from Kielder to Deadwater itself. Its just a sloppy walk, but pretty straightforward.