Yes purchases by debit or credit card can be traced to the location where the transaction was made especially if the transaction was in the store.
As for an online transaction they would have to look up the IP of the person who purchased something or even trace where the package was sent to find out who used the debit or credit card for the purchase.
Your bank can shutdown your debit card for you if someone is using your debit or credit card fraudulently.
They do have ways of tracking and tracing where the purchases were made with your debit or credit card.