How would that really Affect the Exported Animals?
As already stated there are numerous considerations, and a lot of this will also depend on what type of animals you are taking. Let's focus on land creatures first. Magnetic fields, the specific environments/biomes that exist on this planet, the gravitational pull, the cycles of the moon and the seasons, etc. all affect animals. During the voyage a lot of them would likely be deprived of sunlight, their typical territory specifications, and the ability to exert their instincts. Any creature would likely be peeved in that situation.
Also, what are the animals being used for? Farming/domestication? Wild animals?
Aquatic animals I think would be the most challenging, particularly amphibians like frogs that absorb everything through their skin. The chemical composition of a different planet or a spaceship would likely be harder to control for these creatures.
There is ample research to suggest that animals can become depressed, violent, withdrawn, etc. due to long confinement. You can look at documentaries like Blackfish or the Cove and studies such as Learned Helplessness. Birds, especially crows, have demonstrated high levels of cognition.
I bring all that up because, for certain creatures, being confined against their will with many physical changes and no explanation would be almost as distressing for them as for a human. You will need to consider how to illustrate their depressive nature without words (unless they talk or you can hear their thoughts, I guess).
There are plenty of creatures who will be distressed without a high level of cognition, just because things are changing. But I don't think goats or lizards would display that in the same way a crow, dolphin or gorilla would.
I don't think conceptually is something we can really get into without making some decisions about the status of animals' consciousnesses and level of awareness, which I think would vary greatly from animal to animal.
I hope this bit of ramble helps! ;)