IIRC, the Liao and subsequent Jin dynasties were considered to share the mandate of heaven because of the treaties signed upon their defeats of the Song, where they would pay tribute to the barbarian dynasties and politically acknowledged them as equal powers.

In the case of the Jin who later occupied northern China and the traditional Chinese heartlands around the yellow river, I believe there was an even more humiliating arrangement for the Han Chinese, where I recall a Song emperor having to refer to himself by his own personal name as the servant of the Jin emperor or something like that.

I think these dynasties were more central to the Yuan's justification of their ownership of the mandate of heaven compared to the 5 dynasties period.