I don't believe that's how they do it with the money of their clients, banks work on a fractional reserve banking system which means that they only keep a part of the deposited money in the bank while they chip in a lot of it so they can lend it to borrowers, that's why banks can close prematurely during their operating hours especially at a time where a lot of people have withdrawn their money from the bank, they won't be able to accommodate everyone. Another reason why they work is because as I'd hate to say it, banks work on trust ever since, trust that they'll not invalidate you've got money in them by proving it through a piece of paper.
Generally, as you said, but if we as customers are still prioritized and if the supply runs out, they just take it back to their larger head office. There are also moments that create such conditions when many people need fiat in cash, for example during elections or other things, but personally I see that banks have anticipatory steps for important moments like this.