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Probably the greatest natural disaster of the era was the Mississippi floods of 1927. I'm not an expert on the period, but it appears the answer is that it's complicated. Relief efforts were largely dominated by the Red Cross, which had a strong emphasis on local control and volunteer participation. In the context of Missisipi this meant white control, especially white control by the wealthiest families. In this case, it was the Percy family, which employed an immense number of black farmers and sharecroppers. It was a racially unequal process from the start, white victims got more food and better medicine, while blacks were forced to work repairing levees against their will. However, the Percy's also needed black workers to stay, so they had to provide relief, and the growing political clout of black's in northern swing states (where they were largely allowed to vote) meant that politicians across the nation were arguing for stronger disaster response measures.