I used to be a trustee of a local and independent small breastfeeding and baby feeding charity which provides a weekly drop-in session to discuss and seek information/advice with any feeding issues (not just breastfeeding, but BF remains the focus and speciality), or just a safe space to feed your baby and chat. It's a lovely group and well-used locally. While it's not a woman-only space (we welcome fathers if they want to attend - some dads of newborns want to - and there's always a private room to feed in if mothers don't want to feed in front of a man) in practice dads tend not to hang around! Which is fine by us. In practice it's virtually always women-only.
However. Our brief has migrated somewhat over the years, and I do wonder what would happen if, for instance, a transwoman who had adopted brought their baby in. Clearly we want to support the baby, and our brief says we support parents (and mentions mothers in particular). But I don't feel we can or should attempt to provide a support service for a biological man who believes that they can 'breastfeed'. I could see this looming potentially on the horizon and have been troubled about how this would be negotiated - and more importantly, the impact on all our mothers who bring their babies in every week, and are not expecting to see a man in that role. (breastfeeding, not the parenting role)
I left the board for a number of personal and logistic reasons but I still feel very guilty that I've left without asserting what I know to be true - that only biological women can breastfeed, that the group should never attempt to facilitate a biological man to lactate (because that fluid isn't going to be a healthy substance for a baby, because it runs the risk of accommodating breastfeeding fetishists, because it will alienate our mothers). 
There is a big grey area in the mission statements of most small groups like this, I'd imagine, and it can (and will) eventually be exploited by men who want to infiltrate mothers' groups for validation or sheer voyeurism. Yet we do of course want to support fathers and gay parents. It's a bloody minefield. 