Air Traffic Controller reported military UAS started a descent that was not coordinated by adjacent facility; resulting in an airspace violation.

Narrative

I accepted a handoff from RXX on this aircraft descending to 100. Upon talking to the pilot; I asked what altitude they'd like to enter the range at. They replied that they were fine entering at 100; so I coordinated with RXX that Aircraft X was going to stay at 100. As the aircraft neared the boundary of MOA X; they still had not received a clearance onto the range; so I called the range officer to ensure that they were cleared in at 100; which was confirmed by ZZZ Range control. The UAV pilot was talking to the range officer simultaneously; so he did not hear me initially; but on the second try I told him to maintain 100 until established in the range; and terminated his radar service.As he transitioned into MOA Y (070-180); Aircraft X started to descend at a pretty high vertical rate (>1k fpm) all the way down to 020; where the aircraft started to do tight circles over a point. After a few turns; they headed into RXXXX and remained there until I was relieved.Solution: Itinerant military aircraft must have a more thorough briefing of what airspace is or is not hot; and how to remain in their protected airspace unless otherwise coordinated. This is even more critical with UAV ops that do not have the ability to see-and-avoid.In this particular case; my sector was very slow (mostly acting as a frequency relief for [sector] X;) and there was no one under the MOAs for this traffic to conflict with. If there had been a Cherokee or 172 transiting under the MOA as happens very frequently; I wouldn't have been able to get them away from the better-performing-but-effectively-blind UAV in order to prevent an Loss [of separation] or worse.