When you Move any Asset folder by default move option, it will adjust all the references and republish automatically.
And, it is not a best practice to have discrepancy in Author and publisher instance.
What you can do is, create a custom dynamic participant step workflow and keep review the links before actually moving and publishing.
Link:- Adobe Experience Manager Help | Creating a custom Adobe Experience Manager Dynamic Participant step
-Kautuk
Kautuk Sahni