What will become with miners when the issue of bitcoin ends?

They will continue to mine.

The process of mining implies receiving reward not only from the finding of the block but also from transaction processing (at the expense of the commissions). Each transaction has a certain percent (fee) which is listed in favor of the miner who processed the transaction. Thus, in case of reduction of an award for the got bloc at miners still, the motivation will process transactions, including them in new blocks. It is probable that the commission (fee) considerably will increase.

For reference use: The award for production of the block (issue) is reduced twice every 4 years. It is easy to count, issue of bitcoin will continue till 2140.