If your tenants are liable for the council tax and they fall into arrears, any debt will stay in their name. It will not stay with the property or be transferred to you.
When they move out of the property it is important that we are told promptly and provided with a forwarding address if known. This is so that we can close their account and forward all communication to their new address.
Your tenant can not claim CTS if they are not responsible for paying the bill.