Lots of people's views on this topic have evolved as they've learned more. 
Take, for example, tennis legend Martina Navratilova. In late 2018, she entered the public debate about males using their claim of having a trans gender identity to muscle in on female sports when she opined on Twitter: 
“You can’t just proclaim yourself a female and be able to compete against women. There must be some standards, and having a penis and competing as a woman would not fit that standard.”
The legendary tennis champ and longtime vocal advocate for LGBT acceptance and civil rights was promptly pillioried, called a transphobe, and and told to "educate yourself."
In February 2019, Navratilova published a piece in the The Sunday Times (of London) in which she said that she had done as she was told and taken some time to educate herself on the subject - and as a result, she was now even more strongly opposed than ever before to opening up female sports competitions to male athletes who say they have a trans gender identity. Her article appeared under a headline and subhead that said:
The rules on trans athletes reward cheats and punish the innocent
 Letting men compete as women simply if they change their name and take hormones is unfair — no matter how those athletes may throw their weight around
In the article Navratilova wrote, 
I was not prepared for the onslaught that followed [my tweet], chiefly from a Canadian academic and transgender cyclist named Rachel McKinnon [now Veronica Ivy].
McKinnon, whom I had not named (I had no idea who she was at the time)... accused me of being “transphobic” and demanded I delete my tweet and apologise. Since I have spent much of my life fighting injustice, on my own behalf and for others, I was pretty put out, especially when the bullying tweets from McKinnon continued, like incoming fire.
Ever the peacemaker, I promised to keep quiet on the subject until I had properly researched it.
Well, I’ve now done that and, if anything, my views have strengthened.
BTW, I think Navratilova's views about Caster Semenya competing in the women's category have totally changed since the tennis champ published her Times essay in February 2019, several months before the Court of Arbitration for Sport delivered its landmark decision in Semenya's case against the IAAF/WA revealing that the South African runner is XY with a genetic enzyme deficiency, 5-ARD, that causes a DSD only in males. 
More recently, in an interview with The Telegraph last month, Navratilova has said her own views have evolved even further in the six-plus years since then, and they've evolved in a way that has made her more hardline about keeping males out of female sports and spaces. 
I think Navratilova's views about Caster Semenya competing in women's category have totally changed since the tennis 
Key excerpts of what Navratilova told The Telegraph last month:
I went into this with the aim of inclusion. I wanted to include transwomen in women's sports: how do we go about that? 
I became part of a group called the Women's Sports Working Policy Group and with other athletes, former athletes, we were trying to find ways to handicap or level the playing field so that transwomen - males who identify as women, TIMs, uh trans-identified males actually is the correct name for them - could compete [in women's events]. And we found we cannot do it.
And then I kind of got further into the locker room situation rape crisis centers, women's uh refuges, you know refuge, refugee centers for [women who’ve experienced] domestic abuse etc, and I found that in many countries now they're allowing men to go into these spaces. And then I became much more the other way, where we have to exclude them from women’s sports but we also have to exclude males from women’s sex-based spaces.
So I kind of did a 180. I was one of those on the left all inclusive and then I'm like, “Wait a minute. That's not right. It cannot work this way. It cannot work this way.” 
So, um, here we are.
It's mostly men that make the rules and it's mostly, only men that are going into women's spaces -- that's the problem.
[People who support this] are totally discounting women that have been um assaulted or attacked in some way and they’re always putting the feelings of males above those of females. They're much more eager to validate the males who now say they are women than actual women who say “No, no, we don't want this. We need safety. We need dignity, privacy. it's not right, and we're not being listened to.”
For me this whole movement is patriarchy. It’s patriarchy that's that's parading as a human rights issue when in fact it's about males imposing themselves in female spaces. It’s really that simple. 
If you, a man, want to “live as a woman,” please have at it... I will hire you regardless of how you identify if you can do the job. But you cannot go into my uh into my women’s, into female-sex spaces if you're a male.
Women and girls have a right to those spaces.
They say "Oh but it's not fair." Well, what's not fair is male bodies in women's women's spaces. That's what’s not fair.