Even if that's so, the Sixth Circuit could not have "reversed" two U.S. district court rulings in short order like this. Where (and when) was the briefing? U.S. Courts of Appeals sit in panels of three (or sit en banc with the full court) and issue opinions affirming or reversing based on a fully briefed case. That obviously didn't happen here.
As I said, the only thing that could have happened was that a stay was granted. But that's incredibly unlikely, because a stay will be granted only if the full panel is going to hear the case. And we're talking about two separate cases here.
It's time sensitive, that stuff is waived in the interest of immediacy.
I really should have read the whole thing before I spouted off.
While the system usually works as King of Florida described it, the rules change in situations like this, where immediate action (or an immediate stay) is necessary to prevent judicial mischief. What typically happens in such cases is that the appellate court will issue a decision based either on the briefs of the parties below or its own legal research (or, rather, that of the judges' law clerks).