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"Can you normally transfer and use those points on any flight, or does the flight you want have to qualify in some way for you to be able to use points on it?"

It depends on the particular FFP. Some FFPs let you redeem miles on any flight, but the number of miles required can vary from flight to flight. For example, Delta now has five different redemption levels, but the higher redemption levels can require an astronomical number of miles. Other FFPs only offer one type of redemption; such redemptions are only available on flights which have (capacity-controlled) award flights available. The British Airways FFP works that way.