The merchant where you use your card will eventually pay all the fees. In the first instance, the merchant signs a contract with a 'payments acquirer' which specifies the fee they will pay on all different types of transactions (e.g. any combination of Visa/Mastercard/Amex credit/debit/prepaid consumer/business foreign/domestic). Sometimes this payments acquirer will be a bank, sometimes it will be a separate company.
The payments acquirer will then send on some of this fee from the merchant to Visa or Mastercard (again the exact rate depends on the transaction type). Visa or Mastercard will then send on an interchange fee to the bank that issued the card the customer paid with - this is the bit capped at 0.2% for consumer debit cards and 0.3% for consumer credit cards.
So essentially, card payments is a 3 sided market. The merchant chooses which payments acquirer they want to use. The issuing banks choose which card company they want to use. Customers choose which banks they want to bank with.
The interesting thing is that the EU has capped the banks' interchange fee, but not capped the fees elsewhere. When the interchange fee was reduced, this was passed on as a cost saving to the payments acquirer. This saving might have been passed onto the merchant (especially if the merchant is big and sophisticated enough to shop around), but it was equally possible that the payments acquirer could just enjoy the higher profit margin, or that Visa and Mastercard could raise their processing fee.