I'm not understanding this new Bitcoin Core/Classic/Unlimited fork discussion and what happens to your coins and their price if Bitcoin Unlimited comes online.

If you have money at an exchange, I understand that there might be a split where some percentage of your existing Bitcoins gets automatically pushed into BTU.   If I had 10 bitcoins, for example, and 2 got sold and pushed to BTU, both would seemingly be revalued at the new market rate, and your monies would largely be intact (notwithstanding market forces and rise/fall in general).

If you have bitcoins outside of an exchange, in your own wallet(s), what happens then?  If you have 1 bitcoin in a current wallet (Bitcoin Classic?), when you push it back to an exchange, does it get viewed as the Bitcoin Core - or is there a conversion process where it recognizes that the Bitcoin Core is now a new/revalued coin and there is an exchange based on blockchain and your coin(s).  I just don't see any way how having offline coins before a fork is going to work out well.