I think @protist was close with the Night of the Lepus suggestion, but I think a different movie about a bunch of rabbits may actually be more thematically fitting.

Watership Down

A small group of people/rabbits breaking off from a very large group. Forewarning bad things happening in the larger group that they are fleeing. Attempting to invite others to join them along the way to increase their numbers to a sustainable level. Conflict with another group of domineering authoritarians.

It's all there.