Air Traffic Controller reported aircraft failed to hold assigned altitude descending below MVA resulted in a low altitude alert and a CFTT event.

Narrative

Aircraft X was flying toward ZZZ level at 10;000 ft. I was receiving the briefing from another controller when I noticed his aircraft descending without a clearance. I pointed this out to the controller giving him the briefing and the controller verbally told the aircraft to stop his descent at 9;000ft. The aircraft reported struggling to climb back up to 10;000ft. The aircraft leveled off at 9;000ft and I took over the sector. Immediately after taking over; I noticed the aircraft descending again. The MIA (Minimum IFR Altitude) in the area was 8;900 ft and the aircraft had already begun descending below it. I issued a low altitude alert and told the aircraft to climb immediately. About 1.5 miles to the east of the aircraft; the MIA is 8;000 ft. I issued a heading to the aircraft while it was still below the MIA to get the aircraft both into the lower MIA block and to slowly turn him to fly with the wind to gain altitude. The aircraft was able to climb back up to 10;000ft againIn the future I would ask the aircraft if they have the terrain in sight and suggest a heading instead of turning them below the MIA. I believe I did the right thing by issuing a low altitude alert and doing what I could to get the aircraft climbing again.