If you use Mizzium Transreliquat's abilities multiple times in a turn in response to one another, then each time one of those abilities resolves, it will overwrite whatever the permanent was copying. The Transreliquat will wind up as a copy of the artifact targeted by the last ability to resolve. When the turn ends, all instances of its first ability will wear off at the same time. If one of those was the last copy ability to resolve, the Transreliquat will become what it was before those abilities resolved. This will likely be either the original Mizzium Transreliquat or whatever it copied with the last instance of its second ability to resolve.