Frans de Waal, accomplished primate biologist, writes on morality and its precursor or foundation in primate evolution. He argues against what he calls Veneer Theory, the view that morality in humans is just a thin layer on top of an amoral core. Instead, morality, or at least its fundamental building blocks, can be found in social mammals, and in particular in primates such as the chimpanzee. He posits that morality evolved, and evolution does not do discontinuities (saltatory changes). So the development must have been gradual. His views are discussed by the philosophers Robert Wright, Christine M Korsgaard, Philip Kitcher and Peter Singer.
I believe de Waals has a number of valid points, and describes his own and other's findings very well. However, he is rightly criticized by Kitcher and Singer that he is vague when he claims that the "building blocks" of morality are there in chimpanzees, and therefore, by evolutionary extension, also in the common ancestor of humans and chimpanzees. That concept seems to hide more than it reveals.
The basic issue here, I think, is the emergent nature of morality. Emergence is a tricky concept: it says that something new can spring out of something that already exists, given certain context and conditions. But it may also say that this new thing has some kind of beginning, a proto-existence, in the preceding state. And this new thing, when it has come into existence, can be viewed as something "higher up". It is based on what is below, but can also affect what is below. The terms often used, such as "based on", being "the core of" "the foundation of" and similar have a hard time conveying this sense that the emergent property can be both a function of, but also affect, the basis of its existence.
Morality is such an emergent property. It is based on psychological propensities common to humans and higher animals such as chimpanzees, but its reliance in humans on culture makes it different from what exists in chimpanzees. So although de Waal has a point that Veneer Theory is a bad way of describing morality, he too lightly skips over the large differences that culture and language produce when combined with the psychological common traits of humans and chimpanzees. In general, de Waal underplays the importance of culture in humans. Both Peter Singer and Philip Kitcher have valid critical points here.