Works where the main antagonists are "demons"
 

I'm not talking about the typical TV demons, hideous hell-beasts with horns and fangs, or youkai. I'm talking about the more classic sense of demonic spirits, noncorporeal beings that can do next to nothing on their own because they have no physical body, but that have the power to possess and manipulate almost anyone they choose. The less physical form they're able to take the better.
Are there any where one or more of the enemies is able to "travel", like in the American movie Fallen?The concept of an enemy who can be anywhere and anyone, and who can be someone else in an instant's time, would make for an interesting story. On top of that, it can cause a person to be "betrayed" by a loved one, or witnessing a loved one committing some heinous crime, and that loved one may not even remember the even afterward, further complicating the conflict.

Along with a traveller, it'd also be interesting to see one where the possession is more complex and difficult to break; for instance, a form of spirit that burns away the soul of the person it possesses, so that the real person's actually almost guaranteed to be dead, replaced with this other being, and if the possession is abolished then all they'd get back is a hollow shell or corpse.

I'm not looking for something where the possession turns a person into a physical monster. They should be basically unchanged physically.

Directly demons, I doubt there is anything like Fallen in anime. But indirectly a Geass power that brainwashes people, a Death God that gives a Death Note to any person he likes, Johan from Monster, or the Rodan from Tekkaman Blade do that. There is also Parasyte in manga form but the possessed change in appearance a lot.

Supernatural fits the bill perfectly; there are countless demons able to possess human bodies (giving them abilities like inhuman strength or telekinesis) and can keep the host's body alive as long as they're not exorcised or killed by supernatural means. Their only distinguishable feature is a change in the colour of their eyes. Won't go into depth since that would spoil a lot...
Anime/manga wise, the only thing remotely close is D.Gray-man, although demons are programmable machines wearing meatsuits that turn into a mass of guns when they want to kill, so I guess it doesn't count.