What is the ruling on the prayer of a man who prayed Asr with the intention of an obligatory prayer behind someone praying a supererogatory prayer, and then that same person who prayed the supererogatory prayer prayed the obligatory prayer behind the one he had previously followed?

The prayers of all are valid. The first person prayed a fard (obligatory) prayer behind someone performing a nafil (supererogatory) prayer, and this is permissible according to the soundest opinion. There is no harm in the person who entered the mosque joining him, because both are performing fard prayers. There is also no harm in the person who was previously leading the prayer (the one performing a nafil prayer) joining him, because the preponderant opinion is the validity of the prayer of someone who was an Imam for part of it and a follower for another part, and this is even more permissible in this case.