Let me start with a more basic question: If Alice is trying to send funds to Dave,
via:
Alice -> Bob -> Carol -> Dave,
Do Bob and Carol need to actively permission this, or is this supposed to be automatic
as a property of how the LN operates?.
Bob and Carol would need to actively permission this, and each of them would likely charge a fee for going through their open channels. A likely setup would be that both Bob and Carol somehow advertise/broadcast this service, and keep their LN node running, along with their private keys to facilitate this via automation.....in other words keep their money on an online computer.
Block candidate producers (miners) would then be very likely to include the first available transaction that becomes unlocked...
I think it would make more sense if the miners included transactions that pay the highest tx fee rate that are valid. I am not aware of any incentives for the miners to include a transaction immidiately after (relative) time locks expire, nor when a transaction is first seen by the miner.