Hi Zindifir, as you probably know 'Exectuing Auto Move" is the turn indication when AI is working through their turn. When something in the AI behaviour goes funky it will likely present as a hang with this indication. Since it's AI behaviour I think you can understand that there's many things that could cause a failure.
Over Planetfalls lifetime we've fixed plenty of these issues, so when it shows "Executing Auto Move" there's not a single bug with a single fix. We've had AI constantly repeating a fight that would end in a Draw due to an Out of Sync. There's been hangs when AI tries to move on a specific structure at a particular moment. Everytime we fix an instance/occurance, a hidden or deeper issue has the potential to surface.
Generally these issues can be worked around by going back a couple of turns and making different choices. This can result in the AI changing its goals and avoiding the situation that caused the issue.
HHopefully this provides a bit of understanding as to the nature of the Exciting Auto Move/Opponents are Moving bugs.