There is a decent article in NYT that explores the work of Dr. Frans De Waal and his assertion that the root of human morality is grounded in the sociality exhibited by the primates. His contention is that animals (esp Apes and Monkeys) show emotions and empathy; as well as for the evolution of co-operative behavior many other factors underlying Morality- like reciprocity and peace-making evolved and these in turn set up the stage for the evolution of Human morality.
Social living requires empathy, which is especially evident in chimpanzees, as well as ways of bringing internal hostilities to an end. Every species of ape and monkey has its own protocol for reconciliation after fights, Dr. de Waal has found. If two males fail to make up, female chimpanzees will often bring the rivals together, as if sensing that discord makes their community worse off and more vulnerable to attack by neighbors. Or they will head off a fight by taking stones out of the males’ hands.
Macaques and chimpanzees have a sense of social order and rules of expected behavior, mostly to do with the hierarchical natures of their societies, in which each member knows its own place. Young rhesus monkeys learn quickly how to behave, and occasionally get a finger or toe bitten off as punishment. Other primates also have a sense of reciprocity and fairness. They remember who did them favors and who did them wrong. Chimps are more likely to share food with those who have groomed them. Capuchin monkeys show their displeasure if given a smaller reward than a partner receives for performing the same task, like a piece of cucumber instead of a grape.
These four kinds of behavior — empathy, the ability to learn and follow social rules, reciprocity and peacemaking — are the basis of sociality
While it is still contentious as to what extent Morality is inbuilt (genetic and one of the human universals) versus it develops under the influence of society and is culturally determined; the actual case, like everything else, may lie in between in terms of a developmentally unfolding of inherent potentiality and with various nuances as per the culture of flowering. Here Kohlberg's developmental framework would seem relevant- but that framework is too much Kantian in the sense that it emphasizes, and is based on, rational reasoning. The reality may however be as more Humean and as per De Waal and Hauser, whereby most moral decisions are more intuitively guided, with post-hoc reasoning following the initial emotional decision.
But biologists like Dr. de Waal believe reason is generally brought to bear only after a moral decision has been reached. They argue that morality evolved at a time when people lived in small foraging societies and often had to make instant life-or-death decisions, with no time for conscious evaluation of moral choices. The reasoning came afterward as a post hoc justification. “Human behavior derives above all from fast, automated, emotional judgments, and only secondarily from slower conscious processes,” Dr. de Waal writes.
I, of course am most sympathetic to the developmental framework and hope that someone would take up Kholberg's framework and incorporate emotions and emotional intelligence in it.