In fractional reserve banking, the bank is obliged to have a minimum ratio of liquidity it have (to pay for customers' deposits) and the total money its customers have in their bank accounts. But what happens when a customer of bank A transfers its deposits to a customer of bank B? In this case, the bank A's ratio of reserves to deposits increase and will be in a safer state (because it's far from a bank run). On the other hand, bank B would have less reserve to deposits ratio than before the transfer of deposits and thus, is more probable of a bank run. How does banks handle these transfer of money between themselves?
First, your description is not necessarily correct because when banks transfer liabilities (e.g. including customer deposits) something has to happen to asset side of balance sheet. Theoretically that could involve transfer of reserves from bank A to bank B. This means that the description you talk about is not correct.
Second, banks had to abide by reserve requirements before they were set to zero recently, but they did not had to do it from second to second. Banks have constant inflows of reserves (e.g. from payments from other banks) and they simply have to check at the end of the day whether the inflows are larger or smaller than outflows. If they had too many outflows they could (see discussion in this world bank blog):
- borrow from other banks on interbank market
- borrow from central bank (especially post 2009, before that it was possible but heavily discouraged)
- by liquidating some unencumbered assets on the asset side of balance sheet.