New Voices
Mythology gets updated when new voices retell the stories. There are a number of differences between the Lord of the Rings books and the Peter Jackson Lord of the Rings movies, but both are highly regarded by fans.
Similarly, comic book superheros are frequently updated when new artists retell the stories with new tones and different focus.
So to find times when Mythology have changed, look for times when new people have encountered the stories (example: migration) or times when new people have had a change to tell the stories (example: social change).
Hindu Specific
The Indo-Aryan migrations would be an important period in the formation of Hindu mythology. The (presumably well established) Indo-European herding culture mythology would change as it encountered Central Asia farming cultures ~2,200 BC. Wikipedia suggests that the Indo-European Sky Father (Zeus / Odin) was absorbed into an existing Central Asian Indra cult to create the Vedic Indra.
As the migrating Indo-Aryans settled down and merged with existing Punjab cultures, it probably would have undergone another significant shift. This would be around the time that the first Vedic myths were written down, ~1,500 BC.
The introduction of amhisa after the rise of Jainism would be another period of change for Hindu myth (cira 600 BC). This leads to the shift from cow sacrifice to cow veneration. The later rise of and interaction with other world religions - Buddhism and Islam certainly - would also influence Hindu myth, but I'll cut this off here before we get too long.