The glass ter'angreal shows ancestral memories. Rand's ancestors are as follows, in chronological order, assuming "greatfather" means "grandfather":
Charn -> (3 generations) -> Coumin -> Jonai -> Adan -> (?) -> Lewin -> Jeordam -> (?) -> Rhodric -> (?) -> Mandein -> (???) -> Rand.
A couple of the notable aspects of Rand's lineage: Charn served directly under Mierin, and remembers her
drilling the Bore. Mierin is, of course, Lanfear.
Jonai is the one charged by Solinda Sedai to take the caravan.
Their stories are told in reverse chronological order, which makes it kind of hard to follow, but the notable passages to answer your questions come in Adan's and Jonai's stories:
"When had Esole died? So little to be laid in a hastily dug hole, wasted with sickness there was no Aes Sedai to Heal." (Chapter 26)
They then run into a group of Ogier, and Jonai's thoughts wander:
How long since he had seen an Aes Sedai? Just after Alnora [his wife] died. Too late for Alnora. The woman had Healed the sick who still lived, taken some of the sa'angreal, and gone on her way, laughing bitterly when he asked her where there was a place of safety. (Chapter 26)
Notably, in Jonai's first appearance, when he is charged with the caravan of angreal, sa'angreal, and ter'angreal, it describes Adan as being 10 and Esole playing with a doll, so my conclusion is that Aes Sedai did not accompany the caravan.
Subsequently, (or previously), in Adan's memories,
the caravan is attacked. Another member of the caravan, Sulwin, unloads some of the artifacts from the caravan, looking for supplies, claiming the Aes Sedai will never come. In fact, he claims that he and his companions "mean to find a place where we can be safe, and sing again," (Chapter 26). The likely conclusion is that Sulwin's followers become the Tinkers.
Lewin, of course,
is the first to be cast out for killing with a spear, claiming that he is still Aiel. Presumably, he follows the caravan of the Jenn. Some of the Jenn begin to abandon the Way, notably Jeordam's wife Morin, who is the progenitor of the Maidens of the Spear and probably the Wise Ones, as she takes up a spear and mentions seeing Jeordam's face "in the dream."
By the time of Rhodric, the split is even more pronounced:
"We guard the Jenn," Jeordam said. "It is they who travel with Aes Sedai."
The Aes Sedai made Rhodric very nervous, though he kept his face blank. They were only four, not dozens, but enough to make him remember stories that the Aiel had failed the Aes Sedai in some way that no one knew. (Chapter 25)
So some Aes Sedai join up with the caravan at some point, but it is clear that they are very old.
By the time they reach
The Jenn realize that they will not survive, and call the Aiel to them, explaining:
"'Why do you not carry a sword?' That drew angry mutters. 'It is forbidden,' Mandein growled. 'Even Jenn should know that.'
'You do not know why,' Mordaine said, and Narisse added, 'There is too much you do not know. Yet you must know.'" (Chapter 25)
They then charge the Aiel with
Coming to Rhuidean for those who wish to lead to learn why they do not carry swords.
The two ancient Aes Sedai present prophecy of the Dragon's coming and his subsequent destruction of the Aiel, and the Jenn representatives tell them that they must comply, for:
"'Our days dwindle. [...] A day will come when the Jenn are no more, and only you will remain to remember the Aiel. You must remain, or all is for nothing, and lost.' [...] 'It is our purpose,' Dermon replied calmly. 'For long years we searched for this place, and now we prepare it, if not for the purpose we once thought.'"
So they accept their fate, in true Jenn Aiel fashion, as Adan shouts at Lewin,
"'We must accept what comes. Our sufferings are sent to test our faithfulness. We accept and endure!'"
Or as Solinda Sedai tells Jonai:
"'Do you know what happened to the Aiel at Tzora? [...] Ten thousand Aiel linking arms and singing, trying to remind a madman of who they were and who he had been, trying to turn him with their bodies and a song. [...] he stood there, staring as though at a puzzle, killing them, and they kept closing their lines and singing.'"