Center Controller reported an aircraft experiencing a loss of pressurization descended through an active Military Operating Area.

Narrative

MOA active with multiple flights of fighter aircraft. ZZZZ Center was notified to send all traffic via ZZZ. The XX controller noticed the Aircraft X point at ZZZ1 directly into MOA. Data block showed 41000 ft. Assigned but descending through 35000 ft. She called and told them to turn him. They said 'negative; he said [requesting priority handling] descending to FL310' she pvd'd [used the Planned View Display to point out] the aircraft to XY/XZ and called. I walked over to their Controller in Charge (CIC) and sector and ensured they had the information that we weren't sure what this aircraft was going do. Aircraft X checked in with XA requesting to continue descent. She tried to turn him away from the MOA and the pilot said no; this is a priority; loss of pressure in the cockpit. Again we coordinated and XZ/XY was able to clear the fighters out of the way. Although there was no loss of separation it was purely out of luck. The lack of coordination from ZZZZ Center was most definitely an issue. We aren't sure how long they had the knowledge of the [problem]; also they only said he was going to ZZZ2. The proper coordination would have been much sooner; with the information that an aircraft needs to get below 10;000 ft. For pressurization issues and is landing ZZZ. The fact that our fighter aircraft happened to be away from the area Aircraft X crossed was pure luck; this situation could have been very disastrous. Kudos to the controllers for their fast and appropriate response to the situation.