In my view, there are certainly times when more developed nations have a responsibility to take action to create a more equitable world. Moreover, it is also possible to say that it is in the developed nations’ best interests to do so.
Developed nations often do have an obligation to make the world more equitable. The reason for this is that the more developed nations have often developed themselves by, in part at least, exploiting the less developed nations. Many of the more developed nations colonized other countries and then, even after decolonization, maintained a neocolonial relationship with their former colonies. By doing so, they helped to ensure that the less developed countries would have a hard time developing. In such cases, richer countries clearly do have a responsibility to try to reverse the damage they did in the past and make the world a more equitable place.
Of course, nations, like individual people, are not always motivated by what they ought to do. Instead, they often act based on what they believe is in their best interest. I would argue that it is in the best interests of the rich world to create a more equitable world. A more equitable world would like have less conflict as fewer people would be poor and angry. A more equitable world would be more economically dynamic as there would be more demand from people around the world who could now afford more goods and services. In both of these ways, creating a more equitable world would end up helping the developed countries in tangible ways.
In my opinion, yes, they should. Because they are already developed, they should also help other nations in order to achieve an equitable world.