If there was a period of time with no transactions, what would miners have to do?
They would mine empty blocks. Block difficulty would remain the same.

Would they have an incentive to mine blocks? In the year 2140, all mining rewards will be from fees. If there are no transactions, there are no fees. Between now and then, rewards will step down over time. So either:


Clients will keep mining.
Clients will stop mining because there are no transactions.


I recognize that this is a useless answer (maybe they will, maybe they won't) but I hope you understand the difficulty of predicting what mining clients will be like in 5-10 years.