Illaoi's god only demands that life be in motion, as Nagakabouros gives every being their destiny and it would be denying Nagakabouros to deny your destiny. This is shown in Illaoi's color story when she puts Gangplank through his trial. He had been left stagnating as the king of Bilgewater for too long, so he almost didn't make it. His will to rise and become king once more was what put him back in motion to let him pass the trial.
Nagakabouros is all about the flow of life, and life itself is motion, so Nagakabouros can be called a god of life. We see this in Shadow and Fortune when Illaoi straight up annihilates a ton of undead. I would assume that undead are the ultimate form of stagnation.
To help visualize what being in motion is like:
Gangplank was almost stagnating, but he is now in motion.
Miss Fortune is likely in motion, but she might not have had her trial yet. This may become a plot point.
As confirmed by Rioters, Braum would definitely pass a trial, but apparently he would so obviously pass that Illaoi wouldn't bother putting him through a trial.
Jhin would not pass a trial. Someone asked why, as he's pursuing his art and that could be "in motion," to this a Rioter replied:
Is Jhin actually happy?
Is he actually chasing what his heart wants him to do?
There is a bit of subtext in many, many of his lines, here's one on the nose --
"There is nothing for me... but this." --Jhin
If Illaoi could put him on trial, Galio would pass. Apparently divine magic and arcane magic work with different methods, so Galio might be able to get a trial.
Rakan would pass.
All undead would fail automatically, and that might apply to Ascended beings, as they're "trying to cheat death." Not sure if this applies as much to Renekton tho…
Vi would pass.
Now, quotes
Illaoi's religion is about life serving the universe. But Nagakabouros is see as bigger than that goal.
Illaoi knows, she can not comprehend her god or it's plan. So she and her religion are attempting to serve and understand a small portion of that god idea.
Nagakabouros doesn't give out bibles, or hard rules. It demands you to be self aware, emotionally present, and iron willed. Her faith requires you to approach you're own subconcious with simultaneous questioning doubt and total certainty.
Having doubts isn't a "sin" in her religion-- failing to take action is.
Arguably having to test Gangplank, a former lover, was an extremely difficult test to give... but she examined her own motives, listened to her gut, and didn't allow herself to hesitate as soon as she realized he needed to be tested.
Almost all of this information came from this, so if you want the full discussion go there.