Jack Frost is a compilation of Russian fairy tales, but I'll focus on the last part of the film, which deals with the story of Father Frost. Just like Snow White, it was softened for mass consumption (the Wicked Queen is forced to put on red hot shoes and dance til she dies):
A woman had a stepdaughter and a daughter of her own, and she hated her stepdaughter. One day, she ordered her husband to take her out into the winter fields and leave her there, and he obeyed. Father Frost found her there, and she was polite and kind to him, and he gave her a chest full of beautiful things and fine garments. When her stepmother sent her father to bring back her body to be buried, he went, and the dog said that she was coming back beautiful and happy, and despite the bribe of a pancake, went on saying it.
When the stepmother saw what her stepdaughter had brought back, she ordered her husband to bring her own daughter out to the fields. The girl was rude to Father Frost, and he froze her to death. When her husband went out to bring her back, the dog said that she would be buried, and despite the bribe of a pancake, repeated it. When he brought back the body, the old woman wept.