This idea comes up every few months in one group or another. Let's ask unsubscribers why they unsubscribed. Actually... let's not.
Emailing unsubscribers is nothing new, email marketers have been doing this for over a decade but there's always some newbie that thinks they just discovered the biggest secret in the planet and start sharing it and posting about it.
And they automated it whoop dee do!
You may be tempted to put unsubscribers into a new "why did you unsubscribe" funnel. And you can even do it so it looks like you did it manually through your Gmail account, you know, to add that "personal touch."
Hint, it's not a secret and it's not all that impressive. You can easily do this with Zapier, or IFTTT or even manually from within your Gmail account.
Still, don't do it.
If you're faced with the choice to do this or not do it-- Don't do it... yes, it's clever and "neat" but it's annoying AF and most likely illegal, plus it will most likely damage your technical reputation and email reputation with the big email providers.
Should you follow up with people that unsubscribe from your list? Maybe, but probably not automatically. If they are only subscribed to one of your lists and they already chose to unsubscribe, they don't expect yet another email.