Firstly, Hobbits were never mentioned prior to the year 1050 of the Third Age when the Harfoots entered Eriador and in doing so were first mentioned in the records:
1050 The Periannath are first mentioned in records, with the coming of the Harfoots to Eriador.
Appendix B: Tale of Years
The next mention of Hobbits comes in 1150, when the Fallohides entered Eriador and the Stoors move to Dunland and the Angle. This is the first mention of the Stoors in records:
c. 1150 The Fallohides enter Eriador. The Stoors come over the Redhorn Pass and move to the Angle, or to Dunland.
According to the Tale of Years, it is after the slaying of King Argeleb I of Arnor that the Stoors begin their next migration:
1356 King Argeleb I slain in battle with Rhudaur. About this time the Stoors leave the Angle, and some return to Wilderland.
It was at this time that the Stoors that had dwelt in the Angle (between Hoarwell and Loudwater) fled west and south, because of the wars, and the dread of Angmar, and because the land and clime of Eriador, especially in the east, worsened and became unfriendly. Some returned to Wilderland, and dwelt beside the Gladden, becoming a riverside people of fishers.
Appendix A: Annals of the Kings and Rulers, (iii) Eriador, Arnor, and the Heirs of Isildur
It is at this point that the Stoors split. With part of them staying in Dunland and the other group returning to the Wilderland (this seems to be the origins of the Stoors from which Sméagol comes).
It is not for another 200 years before the Stoors are mentioned again, where they join the other Hobbits (who earlier, in 1601, were granted new lands by Argeleb II) beyond the Baranduin:
1601 Many Periannath migrate from Bree, and are granted land beyond Baranduin by Argeleb II.
c. 1630 They are joined by Stoors coming up from Dunland.
At this point the intermixing of the three Hobbit-kinds after this point has led to some difficulty in distinguishing one from another. But those Hobbits with greater Stoorish influence had some linguistic differences, due to the Stoors time in Dunland and with the Dunlendings. Stoors were also the only of the three Hobbit-kinds that could grow facial hair:
But they were well known to be Stoors in a large part of their blood, as indeed was shown by the down that many grew on their chins. No Harfoot or Fallohide had any trace of a beard.
The Fellowship of the Ring, Prologue
Stoors of Wilderland
Not much is known of the Stoors of the Wilderland. They were river folk who sailed in boats and sometimes swam, rare for Hobbits:
I guess they were of hobbit-kind; akin to the fathers of the fathers of the Stoors, for they loved the River, and often swam in it, or made little boats of reeds. There was among them a family of high repute, for it was large and wealthier than most, and it was ruled by a grandmother of the folk, stern and wise in old lore, such as they had. The most inquisitive and curious-minded of that family was called Sméagol.
Fellowship of the Ring, Book I, Chapter 2: The Shadow of the Past
It is also known that Sméagol (then called Gollum) was expelled by his grandmother (the matriarch) for his thieving and mischief:
He took to thieving, and going about muttering to himself, and gurgling in his throat. So they called him Gollum, and cursed him, and told him to go far away; and his grandmother, desiring peace, expelled him from the family and turned him out of her hole.
By the time of the Lord of the Rings, the Stoors of the Wilderlands had long been forgotten and had disappeared from the histories. Their dwellings were deserted and it is unknown if they simply ceased to exist or joined the other Hobbits in the Shire
...no dwelling of Halflings could be discovered in the Vales of Anduin, and that the villages of the Stoors by the Gladden had long been deserted.
Unfinished Tales, Part Three: The Third Age, IV: The Hunt for the Ring