The cons of the movement are that there are a lot of schools of feminism right now, and some of them very strongly disagree with one another. This is actually both a pro and a con at the same time. I think the Aziz Ansari incident was probably the best example of the real clash of worldviews between even feminists on what the movement should be trying to stop. The thing that was most interesting, to my mind, about the Aziz Ansari incident is that there was a lot of disagreement over what consent is, what assault is, who has what obligation to do what, etc. I say this is a pro and a con because it's both. It's important that, as a society, we hash out these issues, which is the pro, but the con is that it starts seriously distracting from really unambiguous and grotesque cases of sexual assault.
Another negative aspect of the movement, which it picked up from the more fanatical elements of the "SJW" movement (for want of a better term), are these kinds of warped notions that come in varying intensities, like this an issue exclusively about women; that women should be the only ones speaking; men should shut up and listen; to confusing personal problems for systemic problems. There's an element to the SJW framework that has real problems with it, because a lot of it started off as very academic jargon and has now moved into conversations that are... not that. So I'm happy people are having conversations (and even more happy that very bad people's power is being destroyed), I just sort of wish we could unshackle ourselves from the more problematic aspects of modern SJW culture while embracing the part about advancing people's rights and challenging illegitimate, harmful institutions. But that's pretty much my only issue with the movement, or at least the other critiques are just derivations on this point (i.e. the people who insist that no one should at all be concerned with false accusations).