Central banks are how banks can move funds to eachother.
Say for example someone with an account at bank1 wanted to send 10 billion dollars to an account at bank2. Bank1 would send funds to the central bank who would then credit funds to bank2 (the more likely scenario is that bank1 would borrow funds from the central bank, or an other bank or would instruct the central bank to debit their account at the central bank). If the central bank did not exist then bank1 would need to send cash to bank2 to facilitate this transfer.
Say for example someone with an account at bank1 wanted to send 10 billion dollars to an account at bank2. Bank1 would send funds to the central bank who would then credit funds to bank2 (the more likely scenario is that bank1 would borrow funds from the central bank, or an other bank or would instruct the central bank to debit their account at the central bank). If the central bank did not exist then bank1 would need to send cash to bank2 to facilitate this transfer.
This is absolute nonce. No one central bank does it. Many banks have so called correspondent accounts between each other. So bank1 has account in bank2 and when client1 send money to client2, bank1 moves money from its account within bank2.
If bank1 and bank2 are rather small and distant and they don't have direct relation then they definitely have correspondent accounts in one of world largest banks (bank3). So bank1 moves money from bank1's account to the similar account of bank2 within bank3.
You are correct when speaking about smaller banks moving money. These banks generally only have a few billion dollars in deposits and assets. The smaller banks essentially use larger banks as a "central bank" however the larger banks like Chase, US bank, B of A, Wells, use the central bank to move funds to each-other. This is also how check clearing works.
you know you want it, right? Except the part where they already issuing currency, which is already mostly in digital form, not in bills..