What might the miners do at this junction?
Obviously there's nothing more to mine, most of them will put a stop to their mining rig.
Might there be a reason to move to non-ASIC hardware?
You just stated the reason, there's no more coins to be mined so no more profit.
Or a reason to change the reward algorithm?
I doubt that this will happen. During 2140, the transaction itself will be scarce. Fees will skyrocket.
I'm pretty sure by this moment in time there will be new technologies that will arise which will be better than bitcoin. We may not be present during those times but I'm positive that the future generation will come up with a better invention that could replace bitcoin as the number one crypto.