I think most people think that miners do just that: "mining coins". But the thing is that their role in the network of a coin is much bigger than that. Their role is to keep the network secure, and by keeping the network secure I mean, processing the transactions and making sure they are all valid. They get the coins and the fees as a reward for doing just that. So "mining" will never stop, because if it stopped then tx were not processed and verified. They will eventually stop receiving the reward when the coin limit is reached, but they will still receive the fees for making the transactions, and that should still be a good incentive. Don't know how things will be when the lightning network gets deployed, since fees would be minimal. But I guess that we would get huge adoption when the LN arrives, so even minimal fees would be very profitable for miners.