How does data actually move behind the scenes when transferring funds within the same bank or to another bank?

Within the same bank it is just a journal entry into the database, crediting one account, whilst debit the other.

Interbank can vary, but it is again dealing with settlement accounts.

Read more about it at:

• Faisal Khan's answer to What exactly happens when a wire transfer is made? How does the money travel from the sender's account to the recipient's account? How much time does each step take? What regulatory measures are in place, if any?
• Faisal Khan's answer to How does the settlement of payments work in banks? Specifically, how do payment systems that are connected to multiple banks actually settle the amount between two banks?