When you deposit physical cash at a bank, the bank accepts the money, and records that in its books to show that your account has a higher balance. The account balance itself is your asset, and the bank's liability. So when you have an account balance of $1000, that's the bank owing you $1000, that they have to pay you at any time, on demand. They can pay it to you with physical currency, if you show up to the bank and say you want to withdraw the balance in cash. Or they can honor your demand to pay someone else on your behalf (through a paper check, electronic funds transfer, etc.).

Like if a bank takes the gym bag full of money, what do they do with the US$1 million

They can hold onto it, or deposit it somewhere else, in exchange for their own balance with some other financial institution.

The overwhelming majority of money in the modern system isn't physical. It's just records in a ledger: paper in the days of paper records, or digital records in the modern banking world. So although physical cash can move around, on the back end, the banks are mostly dealing with each other in updated ledgers, not physically moving physical currency. I might have a credit card account and a checking account with a bank, and I spend with my card in a way that increases the balance of the credit card, and then use my checking account to pay my credit card balance, and then all that gets resolved without any physical currency changing hands. Including with the merchant, who might accept credit card payments electronically, and then pay their employees through electronic payments as well.

Currency exchange is done by supply and demand. A small country with its own small economy and a floating currency is going to be vulnerable to swings in foreign exchange rates, so that country will likely need to hold a lot of reserves to shore up their own currency value, knowing that they have the foreign currency and gold to be able to trade it for their own currency, and keep the supply and demand for their own currency from completely swinging the value back and forth.

So if it's a country that just uses dollars, and someone shows up with a bunch of dollars, their deposit helps shore up that bank's own reserves of physical cash. Or if someone comes in with an electronic transfer of dollars to their account in the bank, that bank now has a higher balance of dollars on its records (recognized by the other financial institutions).

Adding a currency exchange component to this whole thing sounds like an unnecessarily complicated side quest, if you're just trying to tell a story.