A walk on a bank is a slow pull-out of deposited funds which occurs as consumers grow increasingly uneasy about the health of the bank. The end result of a walk on a bank can be financial ruin for the bank, as it runs out of liquidity for investment. This is in contrast with a bank run or run on a bank, which is characterized by a sudden and very rapid mass-withdrawal of funds and investments by investors. Bank runs usually happen when it is obvious that a bank is going to fail, and people want to get their funds out before it is too late.
The events which lead up to a walk on a bank usually start out slowly, and gather momentum with time. Typically, a walk on a bank starts with the early signs of an economic recession, which causes consumers to grow uneasy. As the economic problems begin to be featured more and more prominently in the media, depositors and investors may begin to be extremely concerned, especially if they have deposited funds in excess of the amounts typically covered by insurance. In the United States, for example, deposits are only insured by the Federal Deposit Insurance Corporation for up to $100,000 US Dollars, so consumers with more than this in the bank could lose money if the bank fails.
As a quiet panic grows, depositors slowly withdraw their funds, usually in trickles and spurts, and move their funds to institutions which they think are more trustworthy, or remove the funds from the banking system altogether. Given the large funds which flow in and out of banks every day, it may take some time to recognize a walk on a bank, and by the time a bank is aware of the problem, it may be too late.
Banks with publicly traded shares must disclose information about their profits, so a walk on a bank usually becomes public knowledge within a quarter, when the bank must issue a report admitting that it has lost substantial deposits. This can in turn trigger a run on the bank, as remaining investors and depositors believe that the bank is about to fail, so they pull their funds out. If the bank wasn't failing before the bank run, it certainly will be afterwards.
The motivations behind a walk on a bank are understandable. For people with their life savings in a single bank, an unstable market can be a source of great worry, as the loss of those funds could be devastating. Even big investors and people with diverse portfolios could take a substantial hit if a bank failed. However, a walk on a bank can also ultimately make the problem worse, triggering widespread economic panic and perhaps deepening the problems which led to the withdrawal of deposits in the first place.