Good question. My understanding is that the ECR-20 tokens use the Ethereum infrastructure and are recorded in the Ethereum blockchain and therefore will automatically switch to PoS when ETH 2.0 arrives. Also consider that we mine ETH but can’t mine for example MATIC which is one such ECR-20 token (although some pools may pay out rewards in MATIC you’re not technically mining it, only being paid out in it - important difference).

That’s my understanding, someone correct me if I am wrong.