I believe in a global sense countries do owe the poorer countries. I believe we all have important resources and essentially we are sharing these resources. When a poorer country has a hard time making ends meet than the global community should feel obliged to pitch in and help out. I believe this can be done in financial and non-financial ways.
Yes, a rich country has the resources to provide aid to poorer countries and should do so. Helping the citizens of a poor country is the responsibility of all other countries with resources to spare. Allowing the citizens of a country to die of poor health or malnutrition does not benefit anyone. But helping the citizens of a poor country to get medical attention and nutritious food for their families can help forge a long-lasting relationship between the countries. A country just never knows when an ally could come in handy.
Poverty is commonly referred to as a global issue, and there is no way the rich should sit there and watch as people are starving and dying in other countries in situations where 1st world countries have the ability to aid them. It is our duty to help others in trouble.
Richer countries are morally obligated to provide adequate health care for diseases we can cure and have the means to do so, when the poorer country does not have the means. This often includes education and technology to help bring the poor country up to a level of better health care. It doesn't seem to suit a rich country to deny access to information that should be free, while watching a poor country suffer.
To be honest, many of the less rich countries have as many or more resources, both natural and human, as the more wealthy nations. But the wealthier countries have often exploited their resources and their labor, which has impoverished them and made the rich countries richer. So of course the latter owe something to the former.
There is no moral or ethical basis for one person to mandate or force another person to provide financial aid to a third party. This argument applies to transfers between citizens of one country and even more strongly to mandating one person transferring money/resources etc.. To other people half way around the world. People that I will never meet, that I have nothing in common with, that I likely share no beliefs with and frankly people who are also reproducing at several times our rate which will mean that whatever amount of "aid" is needed today will increase by the same amount of their population growth.
Though not necessarily an argument based on ethics I can't help but point out we wouldn't even know about these people if not for the fact that we're just too damn smart and have invented planes and telecommunications and the Internet. Literally without our inherent superiority, intelligence, ingenuity we wouldn't be able to have this very debate because we wouldn't know these primitive people in far off countries even exist!