This paper by David Weisbach provides a provisional answer. It depends both on how you measure emissions and how you allocate responsibility. The presumption that rich countries caused most damage is generally unsupportable.
Abstract: This paper examines the data on responsibility for climate change due to past emissions. It addresses two aspects of responsibility. First it shows that the data present a mixed picture. By some measures, developed or wealthy countries are responsible for most past emissions while on other means, responsibility is spread widely with poor countries responsible for a majority of emissions. The differences in the measurements are due two factors: whether the data uses a comprehensive measure of emissions and the extent to which the data is aggregated into regions. The more comprehensive the measure and the less aggregation, the more that poor countries are responsible for past emissions. Second, it examines how theories of responsibility apply to the data. The most well developed theories of responsibility that impose an obligation on injurer to make a payment to victims are the theories underlying tort law. The paper shows that standard fault-based tort theories cannot be used to support climate change obligations. Instead, the theory would have to rely on strict liability, give up on the normally required connection between injurer and victim, and accept undesirable distributive consequences. Moreover, it would not be a basis for ongoing obligations to reduce emissions because relative emissions of nations will change over time. Instead, were such a theory of obligation to be sustainable, it could only be used to support a one-time payment for harm.