Originally Posted by
Cal20
I'm no expert but I will try explain it to you.
You have a certain amount of supply based attributes that a family member gets. When in a territory owned by your own faction this never changes, it says as 'fully supplied'. However if you cross into an enemy territory after so many turns (depending how long you are there) your supplies decrease and this is reflected by changes in the family member's attributes i.e. going from fully supplied>rationing>tightened belts> hungry>ravernous. I'm not sure what happens if you stay for many turns after that but I think soldiers start dying. These changes do effect the battle morale of the troops, that is for sure.
If you want to resupply you can either a) cross back into your territory or b) capture that region you are in.
Few more notes. This does not effect the AI factions (I don't think) and it only happens with family members, not captains.
Hope this helped some!