But if you are mining, there are fewer and fewer new coins to be discovered over time, so you have to compute more and more hashes in hopes of finding a new one. My (possibly incorrect) understanding is that processing a transaction 1 year from now will be no more difficult than processing one today, whereas in the future you'll have to mine for longer to have the same chance of getting a BTC.

In other words, doesn't the workload for processing each transaction stay the same over time?