...enters it willingly

I don’t really understand the concept of NPC enters barrier/aura willingly.
What does mean willingly? I understand this phrase as “spend move action to enter the area”, is it right?

Willingly means by choice. They choose to enter it, verses someone forcing them through it.

So if you force move someone into the barrier, the efffects of the barrier don’t happen. IF the person decides to enter with their movement, then it takes effect.

This is more balance than narrative, btw, but can easily be explained with narrative if you try.

How about banes like dominated and moving barrier/ aura? It doesn’t seem to move willingly in the 1St and in 2nd case doesn’t move at all. Should it have 5 sec delay?(whole turn skipped, so they finished in a reach of the boon)

The barrier talks about this, it is “enters willing or ends their turn”, if you move a barrier on top of them, and then they stay in it, they are affected by it.

As far as dominated goes that’s just up to the GM to decide for the game/setting/campaign.