It seems like the actual problem is that it's painful and/or impossible to keep one conversation going across multiple channels. If you solve that it applies to tons of things, maybe starting with customer support conversations.

I'd like a way to just manage "my conversation with Bob" in one place, regardless of choosing email, voice, text, video, multimedia, etc at the moment. Then "Bob" can be swapped out for anyone else, even "Company Inc customer support".