During her talk with Ferane and the other Whites in TGS, ch.16, Egwene elaborates what she will do concerning Rand, once she is Amyrlin and the WT united again:

"If I were me and the Dragon were Rand, I'd know him to be a rational person, for a man — if somewhat bullheaded at times. Well, most of the time. More importantly, I'd know him to be a good man at heart. And so, my next step would be to send sisters to him to offer guidance. [...] Rand al'Thor sees himself as an emperor, [...] And I suppose he is one, now. He will react poorly if he thinks he is being pushed or shoved in any particular direction. If I were to deal with him, I would send a delegation to honor him. [...] A group of three Aes Sedai, led by a Gray, accompanied by a Green and a Blue. He views the Blue favorably because of past associations, and Greens are often perceived as the opposites to Reds, a subtle indication that we are willing to work with him rather than gentle him. A Gray because it would be expected, but also because if a Gray is sent, then it means negotiations, not armies, will follow."

I think Egwene will now soon send her embassy to Rand (probably they will plan to Travel to Cairhien, Tear and Illian till they evnetually find Rand), so that Rand & Co. learn that Egwene has become the Amyrlin of the united WT, that the BA was expelled and Elaida captured by the Seanchan. She might offer a meeting in person in Tar Valon or at some other place. Likely the embassy will also already tell Rand about Verin's revelations, especially the names of certain BA (especially those bonded to Rand like Elza and Nalaene Forrell, but e.g. also that Alviarin, Galina and Sheriam are/were leaders), Aran'gar as the Forsaken among the Rebels, Mesaana in the WT and possibly about other DF.

It's clear that Egwene has to send very capable and experienced AS, whom she can definitely trust. Therefore I suggest that the embassy will consist of Siuan (the Blue), Leane (the Green) and Meidani (the Gray as leader). To send the former Amyrlin and her Keeper would honor Rand (besides, Leane is a charming woman.. ). Egwene also noted how successful Meidani must have been in the past, when she visited her rooms in TGS, ch.12, and saw the many gifts Meidani got during her work as a mediator. Plus, via TAR Egwene could stay in contact with Siuan.

I think this combination of the embassy would makes plot-wise some sense, too. One can expect that the Warders of the AS I mentioned and maybe some additonal soldiers accompany the embassy as an honor guard. This would allow the opportunity to bind yet another Great Captain to Rand, namely Bryne. Plus, some people speculated that Tam might fall in love with Leane. This could soon happen, if Leane is part of the embassy. Finally, once Moiraine is rescued, and if there is a great reunion of the most important characters in Caemlyn, there could be a very touching scene of Siuan seeing her oldest friend Moiraine again, whom she considered to be dead.