On proprietary brake cables, I start with something thin that will clean out the old gunk/grease. Tri-flow, One-shot, or even WD-40 can work for that. Then hang them up until the fluid runs clean out the other end. After they're clean, I use a spray lithium grease - the kind you shake up the can and spray in. I spray in a liberal amount, then hang the cables up again until the grease starts to run out the other side.

If the cable housing has a crack in it, now is the time to fix that as well. I use clear or color-matched heat shrink to bandage the crack.

Cables hanging while the cleaning fluid works through: