Studies on babies suggests that there is a certain morality sense we are birth with. Of course, this instinct must be developed and refined by society. It is, however, unlikely that our moral rules can become really disconnected with this moral instincts.
Most of these instincts are based on compassion. However there are also other emotions that shapes our common (punitive) moral sense. For example Contempt, Anger, and Disgust (CAD) appears as result of perceived violation of moral rules regarding Community, Autonomy, and Divinity (again CAD).
We tend to apply compassion in circles. We are most compassionate with family and friend, then with people in the same neighborhood, same country, same race.... We also extend compassion, to some extent, to other animals - like dogs, cats, horses. We can be compassionate with all the mammals and more. A bird with a broken wing will create compassion in many.
It seems that our level of compassion to another being is directly linked with the level of similarities we perceive about that being. For example bears are children's favorites because they have a flat face and big eyes (citation needed). Dogs have very expressive eyes that moves us...
With this in mind, our natural morality would probably be based on circles of morality and rights. Some people would say that it's OK to eat animals, however it is not OK to provoke unnecessary suffering. Other would argue that we should only eat vegetables and not animals. The circles might vary, I heard about people that loves their pets more than other humans.
Can morality be really absolute, not subjective and not centered on human? Can we apply the same moral also for animals? I would say that probably not, and here are the difficulties:
Let's say we want to apply our moral to all the mammals. Therefore, we should refrain to eat other mammals like sheep, beef. Cow and pork are already excluded in certain religions - because of different reasons. If the humans would not be able to be fully vegetarians, we can still eat chicken. Would that... fly?
Hmm, what will we do with the other mammals that eat... other mammals? Should we force wolves and lions to only eat chicken? What if we think all animals with brain deserved protection by moral, what will we do with the other animals with brain that are adapted to eat them? Should we force wolves and lions to starve to death because it is immoral to eat another brained animal?
There is also the ecological equilibrium. Let's imagine a morality for rats. We don't eat rats, but is it moral to kill them? Well, they are mammals, some have a metabolism very near to us, we use to test new drugs on them. Should we let them flourish in our cities? Should we feed them if they starve?
We, humans, are actually part of the ecological equilibrium. If we try to retreat from this role, the ecological imbalance might get even worse. What will we do with cows? Maybe we will find it immoral to use the milk, but they are already adapted for this life. AFAIK, without proper milking a today's cow might even die. Anyway, without humans they might not survive in the wild.
If there is something about morality that we can apply to animals, it should be in the first place with the animals that we have tamed. These animals are already very dependent to humans that does deserve some human responsibility. You cannot let a city dog in the wild, he is no longer adapted as the wolf ancestor was.
We have some responsibility to respect that silent contract that we closed thousands of years ago when we tamed these creatures: we help them live and they are providing us with services (dogs, cats), byproducts (eggs, milk) and even their flesh ultimately. In our defense, their species might not be so widely spread today without this "relation" with humans. Is it a fair exchange? I don't know.
I think that our current morality was created by humans, for humans to flourish. I foresee big consistency issues in extending it "as is" to other animals, in the condition of limited natural resources. However, there are some good news. Some of the human morality naturally extends to other species.
We don't want cruel people in society, so we will not accept unneeded cruelties to animals. If we want to consider "morality derived from space colonization" we might even conclude that the best way to serve humanity is to protect a level of diversity and well being for most of the species, including plants - minus some viruses.
A big part of human welfare is connected to the welfare of other species. This creates a reason to try to extend some of our compassion to other species. To apply the full human morality on other species?... I would not bet on this...
One thing that could give humans some higher ground to... "exploit" other species is this: humans are the only specie that can hopefully save these other species in the event of the kind that destroyed dinosaurs. It is a very tough contract, but humans have monopoly on this for now...
This also comes with a huge responsibility for humans. We are responsible to save the life on Earth before being destroyed by a cataclysm. Of course, I presume that we will manage to not destroy it ourselves...