As long as they are mining it they do. Only in infinitely diminishing amounts. Once a miner has transacted with their mined coin (to anyone but themselves), it has been issued. Over 90% of Bitcoin has mined, and I imagine most of it has been distributed as well.
Discussion
This isn’t the right way of thinking about it. The miner will always spend the bitcoin for something that is worth it. So bitcoin puts a bounty out in the physical world to provide something that is worth it. Once the network issues the bitcoin to the miner, it is issued and the bounty is created.