Individuals of a species can either get new mutations in an area and stay there or then move to a new place.
In either case, their new mutations could be good, bad, or indifferent to their survival.
In some extreme cases were there is an obviously new and better way of doing things (like animals breathing air) individuals carrying that mutation might seek out places (out of the water when a large predator approaches) where they could take advantage of their trait.
Undergoing a long (and apparently new) migration to take advantage of some new feature would require knowledge by the animal of its environment, beyond which is reasonably expected (knowledge of where to go when you have not been there before) of the animal's environmental awareness.
Genetics of a population (meaning its evolution, resulting in changing abilities, such as photosynthesis) can change the environment (oxygenation of the atmosphere is a good example).
Alternatively, environmental changes can affect a populations evolution by selecting for genes better suited to the new environment.
It is also possible for changing environments to result in migrations of populations to places more like their former environment.
This would be like a species in a warming climate zone.
It could move a short distance and increase its altitude to get to cooler temperatures.
Small positional changes in distance are much more likely than longer ones, but some species's can move farther than others.
In most cases, the mutation rate and kinds of mutations could be assumed to be unchanged.
The different results, in different environments, can be attributed to different selection pressures on the population's genetics.