For some background. To get an injunction, a court order to do or not do something, you generally have to prove the following:
Likelihood of success on the merits, ie you're going to win at trial, most likely
Irreparable harm, something that can't be fixed after the fact, as by an award of damages
The public interest
The harm of any injunction outweighs the harm of the action enjoined
So, when considering a stay of an injunction, an appeals court reviews these things, often focusing mostly on the first two. The last two things tend to kind of merge, except in purely private lawsuits, like say a patent infringement case.
When the party is the government, it tends to be rather assumed that what's at issue favors the public interest, that is, that what the government is doing is in the public interest and enjoining the government is not. But the last factor can undo that, as in the case of when the government is attempting something clearly or even marginally unconstitutional.
The grant of injunctive relief by a trial court is usually "fact intensive," like Sara Ellis in Chicago finding Bovino and fICE to be full of shit, or in the redistricting case that Texas' motivation was more racial than political.
Appeals courts and SCOTUS especially, should defer to the trial court's evaluation of factual issues, like credibility. Thus, the usual way to reverse (on full decision on the merits) or temporarily stay (shadow docket) is to find that the trial court applied the wrong law or misapplied correct law.
The shadow docket is almost always reviewing an injunction and applying some form of the above analysis. Two places they have gone horrendously wrong in the pro-Trump orders are:
when evaluating likelihood of success, they apply law they think they're going to make, not the law as it stands (and then they get mad at the lower courts for following the law as it stands without ever having given an opinion that changes the law or explains how the new law should be applied)
their evaluation of the harm, like Kavanaugh's stops that ignore that people are imprisoned for days, weeks, or months on flimsy and unconstitutional premises
From a sort of political science or governing perspective, changes in the law should come slowly or gradually especially when not made by the legislative branch. By doing the above, the Court is making abrupt changes in the law, or permitting the executive to do so, without due consideration for the status quo ante, the way things are, have mostly been, and in most cases should stay that way.