It is pretty clear to me that female circumcision is barbaric. The Muslim restrictions on women and their daily lives (like driving and requiring segregation in the workplace are not quite barbaric, but they are foolish from any economic or pragmatic viewpoint. Dowries and arranged marriages and marriages restricted to castes and subcastes are not something that I think Americans understand well enough to oppose formally. I see no problem with Indian or Muslim women (or men) opposing it.
What does any person owe to another person in a different culture? Do we have some obligation to encourage or support religious movements in China?
All people are limited in their powers by the laws, mores and customs of the countries are societies in which they live. Americans and Westerners have customs they would not violate (crossdressing, for one) but we do not even consider these as restrictions. To an Indian woman, a dowry is totally normal. There are probably a huge number of women in Saudi Arabia that have no desire to drive, and who would feel very uncomfortable to leave the house without a male relative escort.