In Harper Lee's novel To Kill a Mockingbird we find Calpurnia taking Atticus Finch's children to her church First Purchase Presbyterian Church: A mainly-black church to which only the black citizens of Maycomb attend service.
In order to answer your question, it would be interesting to identify exactly what you refer to as "the black cause". To a casual reader, "a cause" is something for which people fight and for which people get together to battle or defend. In the case of Lula, her only action in the novel is to demonstrate that prejudice can go both ways: From whites to blacks, and vice versa. This is the case because Lula questions Calpurnia as to why she brings the white Finch children to their church, when they "have their own church" that they can go to.
That is merely bad attitude, and not a defense for any cause. In fact, so bad is her attitude that she does not even check herself before saying something so rude before children. Hence, Lula does not defend any cause. She is acting no different than the white people at Maycomb.
Calpurnia is not defending any cause either. She is there because she wants to go to her church and attend Sunday service like a good churchgoer does. She takes the children because she wants them there with her, since they are like her family. She even reminds Lula how we are all the same and that being rude to the children will not fix the situation at Maycomb.
Hence, Lula is just being proud and Calpurnia is being firm. Both women have reasons to act the way that they do, but there is really not a cause that they are defending. If anything, Cal is defending the right of anybody to go and worship wherever they feel that is worth doing it.