IIRC, Bitcoin transaction fees aren't a fixed percentage or amount, they are something included as part of the transaction that is essentially a bounty for the miner to verify the transaction, and miners can preferentially decide which transactions to verify first based on the offered fees.

Presumably, as creating new coins stops being a source of reward for mining, the cost you need to offer with a transaction to get it verified will (assuming bitcoin remains in active use) go up.