Whats up with the white cloak way of light faction? I mean, its clear they dont like the magic using women, at all. Though, where does their territory exist? I mean, the tower is where the mage women live right? The white cloaks can just camp outside the city?
According to the books, Whitecloaks are based in the kingdom of Amadicia, where they rule in all but name. The Children, however, come from everywhere and are sent everywhere. In the books, when the Whitecloaks are camped outside of Tar Valon at this point in the story, they're not breaking any laws and are just, in general, being nuisances. The way they operate is to encourage others to name people darkfriend so the Children have cause to hang/burn/kill them. There's little protection under law for citizens in any country, and being a darkfriend is a capital offense everywhere, so the Children are able to operate as essentially mostly-legal vigilantes. Plus, they're capable soliders, well drilled, trained, and equipped. So, as long as they're staying within the lines, most countries will not deal with the problems of evicting the Children. Of course, if the Children think they can get away with it, they'd gladly kill some more Tar Valon witches and hide the bodies.
What's missed in the moustache twirling villainy of the show Children is that many normal people have a positive view of the Children. They root out dangerous criminals, and preach a good line. Its why they have so many numbers because they can recruit well anywhere they go. So there would be substantial political capital spent in evicting or barring the Children. Plus, doing so means that the Children are almost guaranteed to name that ruler darkfriend or say that they are aiding darkfriends. Why else turn from the Children of the Light? This increases the double-edged sword for most rulers. That and the Children, being a militant organization, can actually cause significant problems if a force is massed and sent back into the country to root out that corruption. They are an army.
Tar Valon has the additional problem of everyone hating them for the most part. Aes Sedai are not well liked by most. The Children do a lot of encouraging the negative stereotypes but it's not just them and the Aes Sedai themselves have earned a decent bit of the reputation honestly. If Tar Valon tries to exile the Children for anything less than an obvious bad act, then it would give the Children enough cover with the other nations to mount a full on siege of the Tower.
Not sure how much of this translates to the show, though. I stopped on ep 3. The depiction of the Children and the general level of storytelling (mostly Tell don't Show and treating the audience as if they're largely incapable of understanding nuance or complicated concepts) would lead me to believe they don't at all and it's pretty hard to reconcile how the moustache twirlers could possibly be allowed to camp inside such a powerful nation as Tar Valon.