Why Lanfear Had To Go
In News Comes to Cairhien, Moiraine tells Egwene and Aviendha that tomorrow will be difficult for them all. This is true. Moiraine is flung around by Lanfear, whom she attacks while dangerously hurtling into another world, and the girls are injured by Lanfear, Egwene grievously.
At the docks Moiraine is dressed in her best clothes and jewellery, not only her kesiera, but a necklace of large sapphires and a silk dress shot with gold thread. Since she was at pains to duplicate what she saw in the rings, placing the bracelet angreal precisely as she remembered, her expensive attire may also be relevant. Is it something to pay the Eelfinn when they bargain?
The rings at Rhuidean showed Moiraine this future and two others: one where Lanfear kills Rand, and the other where Lanfear enslaves him as Lews Therin. Lanfear must be removed since she knows Rand has the access keys and believes Rand is beginning to consider her idea for them to use the Choedan Kal together (The Fires of Heaven, Gateways). She had set up a confrontation with Rand and Sammael which supposedly she, Rahvin and Graendal would use to take Rand (The Fires of Heaven, Prologue and A Silver Arrow). She could present this to Rand as an opportunity for the two of them to kill three of the other Forsaken in one blow with the sa'angreal plural. Whatever her ideas about Rand’s capture, soon she would have pressured him to use the Choedan Kal with her. If he refused to share them with her, she would kill him, or, if she is able to Compel him, then he becomes her slave and they use them together. This is why she had to be removed.
Moiraine does not know anything of the future that comes after she enters the doorway except for one small thing that doesn’t concern Rand, or Lan either. My guess is that it concerns Thom and perhaps indirectly her rescue party.
Moiraine embraces saidar and leaps at Lanfear to push them both through the ter’angreal doorway face to face. Moiraine claws the bracelet away and Lanfear may be burned out as the angreal is ripped from her while she is drawing so much saidar through it. After buzzing a bit, the doorway burned (see Doorway Ter’angreal article). At a booksigning, Jordan explained that the doorway burned in part because both were channelling and the world on the other side of the doorway has a radically different set of natural laws. (Yet channelling is possible in the world of the Aelfinn; after all, Rand made a sword of fire without ill effects when he met them.)
The realm of the Eelfinn is on a different plane of existence from the main world. Moiraine and Lanfear may have torn the connection between the worlds, thus melting the doorway and breaking the Warder bond. A ter’angreal melts if it is overloaded, or misused, or is unable to connect to what it is linked to. For example, the female access key melted after its sa’angreal disintegrated (see Broken Ter’angreal article). The connection between the land of the Eelfinn and the main world was probably snapped when the doorway ter'angreal was destroyed, snapping Moiraine’s bond to Lan as well. Lan could not feel Moiraine anymore, but he could feel Myrelle, and assumed Moiraine was dead. The ter'angreal doorway in Tear connects to a different realm (of the Aelfinn), and the Tower of Ghenjei is closed unless the right mark is made in the right way, so these connections may not have been able to keep Moiraine’s bond to Lan operational when the connection to the Eelfinn’s realm broke. I think that without an open connection between the worlds, the bond breaks. Therefore Lan’s bond transferred to Myrelle because Moiraine was effectively dead as far as the bond was concerned.
I think of the Aelfinn’s world as a tangential world, touching the main WOT universe at only a few points. If you pass through one of these points, such as one of the redstone doorways, you pass through a singularity. So you can't see what happens to you in there, or whether you come back or not, and you certainly can’t see what happens to you after you come back to this world. If you come back.