Once the active player uncovers a locked die icon and places the appropriate die (or dice) on it to lock it, does this remain locked all through the other, following players turns until it the task/monster is complete? Or do you remove/unlock the locked die right after the active players turn ends, making it freshly unlocked for the next player?
From page 11 of the Rulebook.
Cards and Markers with a Locked Die Icon
Until that Adventure card is resolved, the Mythos card is no longer in effect, or the task on that Monster marker is complete, that die may not be added to die rolls and its result may not be used to complete tasks.
So the dice remains locked until one of these three conditions are meet.
(i) Until that Adventure card is resolved,
(ii) the Mythos card is no longer in effect,
(iii) the task on that Monster marker is complete.
Just ending your turn isn't one of the conditions so it remains locked on following turns unless one of the above occurs.