When all bitcoins get mined, miners will do it for transaction fee's.
So the miners will mine for the little fees? I don't think anybody will do such thing unless the fees are increased a lot more than what they are now which is going to harm bitcoin considering that the price will probably have reached to the moon when all the coins are mined.
That's the idear, yes... When all bitcoins are mined, an equilibrium will be reached.
A lot of miners will stop mining because it's unprofitable for them to mine for those fees, lowering the diff, making sure the leftover miners will find more blocks with the same hashrate, thus collect more fees.
This is a process that will start long, very long before the block reward will go to zero tough.
It's 12.5 now, but only 6.25 in less than 4 years. 3.125 in less than 8 years, 1.5625 in less than 12 years, 0.78125 in less than 16 years.
https://www.smartbit.com.au/charts/transaction-fees-per-block tells us the current fee is about 0.5
BTC per block and rising, so i guess in about 12 years a miner would get at least half of his income by collecting fees and half of his income by the block reward...