If I can argue this by example... from some women I know, there was sympathy with Rowling when she criticised the term "people who menstruate" used instead of "women". Menstruation is a fundamental experience of nearly all women, which has historically been heavily stigmatised and to an extent still is, and has been heavily under-considered in terms of health and hygeine. It feels to them like the experiences and reality of the vast majority of women, that they have fought for for a long time to be improved upon, are being set aside or stripped from them. A feeling that for women's organisations to discuss menstruation as a women's issue could be attacked as exclusionary: leaving them again feeling unable to discuss it freely or think of it as a woman's experience. I think for many this is a challenge and a frustration; for older ones particularly because it intrudes on closely-held beliefs they spent a long time struggling over, and like anyone, letting go of such beliefs is not easy.I did mean in the former sense because so often the argument seems to be framed around changing room situations (for example). I mean, what real threat is there to the identity of cis women from the extremely small numbers of trans women?
And to be fair, in terms of more explicit threat, I don't think a lot of these women are so much worried by trans women in their gender-specific spaces as they are worried about men pretending to be trans women in their gender-specific spaces.