When I play Temporary Lockdown against my opponent, the only permanent of my opponent's that will visually appear to be exiled 'under' Temporary Lockdown is the first nonland permanent they have played to not yet be removed. Other permanents they control which have entered the battlefield more recently will still appear to be on the battlefield, but will be inactive and cannot be interacted with (I became aware of this when I cast an edict spell against one of my opponent's inactive creatures and it resolved immediately with the creature remaining on the battlefield). Creatures will repeatedly appear to have summoning sickness despite having been on the battlefield for multiple turns. This bug has been seen over multiple games by myself. I have not seen this bug from the perspective of the person having their permanents exiled by Temporary Lockdown.