It's great that you can set a ticket to not re-open when a customer replies after a certain amount of days.
On the new ticket it says:
A new ticket is created on behalf of this Ticket
Nothing is commented on the old ticket.
I suggest it would be helpful on the new ticket to say something like:
New ticket created - email@domain.com replied to Ticket #257 (hyperlinked) - older than xx days
In addition it would be great to add a comment to the old (closed) ticket saying a similar thing:
New ticket created - email@domain.com replied to this ticket after xx days, so new Ticket # 4567 (hyperlinked) was created.