If Mosskit hadn't frozen to death, she would of made it to Riverclan with her litter mates safely, and grown up a Warrior of Riverclan.
Only Mosskit had strange memories while growing up, memories of unfamiliar faces and another she-cat as their mother. However these memories were fuzzy at best and her father Oakheart constantly told her it was just a dream. So she eventually pushed these thoughts aside.
Growing up in Riverclan wasn't east for her either, she didn't like swimming or water in general (getting her teased for being a "drypaw") and she much preferred catching and eating voles and birds to mackerels and freshwater carp.
She eventually got her name as Warrior for being a strong fighter and hunter (even if she wasn't a great swimmer like her siblings.). Her name was Mossheart. Oakheart asked Crookedstar to give her that name, as Oakheart's heart ached whenever he looked at her, as she was so much like her mother Bluestar.
Once she learns about Bluestar being her mother, at first she was disgusted as were her siblings, but she realized it made sense why she couldn't fit in. And when Tigerstar took over Riverclan, Mossheart fled to Thunderclan and begged Firestar and Graystripe to save her siblings and Graystripe's kits.
After the battle with Scourge, she in fact decides to stay with Thunderclan, feeling its what her parents would have wanted: Her to fit in.
She grew older and older in Thunderclan, making the great journey with everyone, and eventually taking a mate (no one knows who it was, its thought to be either be Tornear of Windclan or Blackclaw of Riverclan. As she grew close to them both after the journey). She later has a kit: Stormkit, who grows into a warrior named Stormbreeze.
She later dies of greencough around Long Shadows time, when all the cats were moved to the old twoleg nest.