The influence goes both ways. Cultures influence religion, and religion influences culture.

Originally some societies practiced FGM, then when Islam came along, given how well the practice fits in with its obsession with female modesty and conception of women's role, it adopted it, which helped it spread further, including to non-Muslim minorities.

Once something becomes the norm in a majority-Muslim society, there is cultural pressure to adopt it by others.

After all, isn't that we were are counting on in the West, that somehow Muslims will adopt our norms on the matter?