From one obvious angle, the argument against the Libertarian position that all true rights are 'negative rights', this is fully redundant. There are fundamental duties whenever there are fundamental rights, and handling one or the other thoroughly enough automatically covers both.
A 'positive right' creates duties for others to fulfill it. And a 'negative right' creates the positive right that it be enforced, which creates duties for others to fulfill it... (Unless you are a hard-core Libertarian, in which case a negative right creates no other rights. But that means basic human expectations like safety and fairness are luxuries.)
From that POV, all discussions of rights are really discussions of duties. When we discuss the rights of a child, we are discussing the obligations of parents, or the obligation of the state to presume loco parentis and find other adults who will fulfill those breached duties. When we discuss the right to healthcare, we are discussing the obligations of those with medical training...
We would be wiser to speak more openly about the unfulfilled mandates that our thinking about rights automatically conjures up. For instance, as H. Ross Perot harped on endlessly, the U.S. government is a risk of being obligated to spend two or three times its GDP instantly, if everything our laws are obligated to prevent went wrong at once. This is a side effect of our speaking mainly of rights, but enforcing primarily the duty to act according to stated rules. The two don't logically fall before the same sets of decision-makers, with goals and resource limits set in legislation and execution left to policy bodies. So ends up being difficult and confusing to do the math.
But only from a very narrow political philosophy is anything actually missing here.
As for the basis for framing such duties, the ethical and political thinker that has most significantly considered duty as an organizing principle has been Kant, and his work has most recently been productively extended by Rawls.
From Kant and Rawls ethics, oversimplifying slightly, we are all obligated to meet the requirements we could always all expect from one another to the degree we can universally hold the expectations and find them functional and fair. Other than the degree of complexity and the obvious dangers that tradition introduces into the legislating of morality, there is no good reason why the political basis of duty should differ significantly from the ethical one.