I think that the Great Depression did a fairly good job of repressing the politics of identity and difference in the 1930s. Simply put, everyone was broke. Everyone suffered. The emergence of identity that was seen in the 1920s was severely put on hold while the issues of economic relief, recovery, and reform drove domestic policy. Many groups who were vying for establishing a political identity based on differences had a rather complex relationship with Roosevelt and the New Deal. On one had, they appreciated that the President was willing to help all Americans. FDR had done a very good job of essentially convincing America that collective success and identity was the only plausible path to be pursued with something so large as the Great Depression. Yet, there was not a real acknowledgement of the need to politically, legally, and socially advance the causes of identity- based politics. Langston Hughes has a great poem on this entitled, "Waitin' On Roosevelt." In it, Hughes argues that the promises of collectivity are wonderful, but they do not speak to African- Americans and people of color, who were hurting far worse and earlier than the rest of the nation. Indeed, the invocation of American solidarity and collective identity did a very good job in quelling any legitimate political threat to Roosevelt, which ended up being part of the reason why he was able to do what he did without much in way of Congressional opposition and loss of political capital.
Political tensions and the causes you mention of women's, African-American rights and child labor concerns are largely separate in the 1930s. There were some advances, particularly for blacks with the help of First Lady Eleanor Roosevelt and her husband FDR, along with the first woman cabinet member Frances Perkins, who was Secretary of Labor.
But in general, the widespread economic dislocation and desperation brought on by the Great Depression shunted these movements to the side as the focus of both government and society became how to survive and recover from this disaster. Children, like their parents, considered themselves lucky to get the work and stave off hunger.
There were other political tensions, between conservatives who thought FDR was ruining the country with large social programs like Social Security and the WPA, while the Communist Party grew in numbers during this time too, as leftists lost faith in the capitalist system altogether. Huey Long, a Governor and then Senator from Louisiana, proposed a "Share the Wealth" program to compete with FDR's New Deal, and Father Coughlin became the Rush Limbaugh of the 1930s, berating the government, foreigners and Jews.
Lastly, you had people like Charles Lindbergh, the famous aviator who, along with a group called the German-American Bund, sympathized with fascism, Hitler and the Nazi Party.