The constant US dollar price series preserve the growth rates exhibited in the constant local price series. They are not derived using the US price deflator.
The index is created by dividing each year of the constant local price series by its base year value (e.g. 2010).
Then each year's index result is multiplied by the corresponding current US dollar price value as shown in the base year (2010). The dollar figures can be converted into local currencies using the base year's (2010) official exchange rates.
World Bank: For a few countries where the official exchange rate does not reflect the rate effectively applied to actual foreign exchange transactions, an alternative conversion factor is used. It is possible that a country has reported values for GDP in local current prices, but not constant ones - or constant prices but not current ones. This may be the reason for breaks in series or obviously unrealistic growth rates - or growth estimates are available, but actual current price data are not. Constant US dollar series may also not be possible, if local price series are missing for the base year.