You can use this to increase the fee of a transaction after the fact. So, suppose A has a fee of 0.001 in the first transaction, and the current fee needs to be 0.01 . Then, B has this transaction, but no-one will put it into a block, so B hasn't really received his money yet. A could sign another transaction with a higher fee that would send the money back to A. This could happen until the original transaction was actually deep enough in the chain.
But using your situation, B can do the same thing as A. B can create a transaction sending his own 0.999 BTC to another address of his, and include a fee here of 0.019 . In this fee, there is 0.01 to fund this transaction, and another 0.009 to fund the fee A forgot to include. Miners will see this, and include both transactions, and B is guaranteed he has his money.