At what point and in which places does a currency’s value change—at money changers, in banks, or elsewhere?

If you are referring to exchange rates as seen at Money Changers (or MTOs) and Banks, it is because these values are provided by the treasury department of the banks. They set the rate a currency paid (say US Dollar - Euro) after reviewing the rates from the various FX markets and the markets from which the banks Buy/Sell these currency pairs.

The treasure department of a bank would be responsible for looking at the rates, margins, risk and then quoting a figure which they believe would be safe enough for say a 24-hour period when the next day's rate would be published.

The rate is essentially factored by the Demand/Supply of the currency pair in the market.