You define a consensus rule in NewStakeChain that it must include within it a valid block for OldChain that contains no transactions, or otherwise causes damage to the network by making it unusable such as a reorganization proof. Once the OldChain network has been destroyed by the new one, the consensus rule can be relaxed and there no longer needs to be POW completed by anybody.
This specific example works and is economically rational because miners could make a PoW block and get its reward, or they could make the no transaction attack block and get its reward, plus the reward on NewStakeChain for submitting the proof.
Nobody would be mining OldChain so it's difficulty would be irrelevantly close to zero, you'd just just drop the requirement for POW down to something effectively gone. I said it's contrived as anything, but it's all technically possible if you mess with the incentives and intentionally create really perverse systems. That's the messiest of them though, changing the transaction format, script language, issuing new coins or whatever is pretty clean in comparison.
> Nobody would be mining OldChain so it's difficulty would be irrelevantly close to zero, you'd just just drop the requirement for POW down to something effectively gone.
Why would nobody be mining OldChain? You can't remove the block reward or change the difficulty without a hard fork. PoW miners would still decide which blocks are mined and reap the rewards (block reward and transaction fees). You haven't actually proposed a solution that wouldn't require a hard fork.
You can remove the block reward as a soft fork, the consensus rule for Bitcoin and its derivatives is a maximum. Even if it wasn't, you can soft fork in a rule that the block reward should be spent to somewhere invalid, or paid to a particular person.
That doesn't work, unless 100% of miners accept your rule change forever (aka impossible). It would only take a single miner to keep mining using the old rules and this miner would reap the block reward / transaction fees. An old Bitcoin client would follow the chain created by this miner, wouldn't it?
This specific example works and is economically rational because miners could make a PoW block and get its reward, or they could make the no transaction attack block and get its reward, plus the reward on NewStakeChain for submitting the proof.