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Scores do change frequently and often more than one pull will be done during the purchase process. It seems rather common to have an additional report pulled right before closing to make sure nothing has changed dramatically that would disqualify you such as new tradelines, collections, late payments, etc.
They will use your middle score but they will also use information from all 3 reports. For example, if your middle score is Experian but you have a collection listed on Trans Union... that collection still counts against you.