The new owner should reply to the customer. If the new owner does not do this, and the ticket is handed off multiple times, does the customer need to know it?
"Hi, I'm taking care of your ticket"
vs:
"Your ticket has been taken by Bob."
"Your ticket has been taken by Fred."
"Your ticket has been taken by Edward."
I realize you want to do this "automatically" but until the owner interacts with the customer, the customer should safely assume that the ticket is being addressed and need not be involved in how your agents handle it.
Personal opinion, yours may vary.