Would the users using the files be making changes to the already existing rows?

Or maybe we could put in a code that states that as soon as a particular column is updated in the sheet, that entire row should be copied to the master file. But this code would need to be copied to each user's sheet.

Another alternative is to clear out the master file each time you run the macro and start loading it with data from the topmost row of each file. This way, there wont be duplicates incase the users happen to update any old data.