The question of trans people in sports has been something I've thought about for a long time, but I can never come up with a good argument as to why they should be allowed to compete with the gender they identify with post-transition. I have issues with both sides of the argument though, so if anyone can give a good argument to me, I'd love to hear it.
On one hand, I realize that for MtF people, having years to train as a male with all of the physiological benefits that provides will carry over at least somewhat even after transitioning. I think it's similar to how being able to train while on PEDs will offer benefits even after you've stopped using them, though not exactly the same. Trans people (I believe) still retain things like a higher lung capacity and a higher bone density even post-transition. These differences make me think they should not be allowed to compete against the sex they transitioned to because someone born as a woman would never have the chance to train with those advantages.
On the other hand, FtM people often take testosterone in order to transition, and if any other athlete was doing that, I think everyone would consider it doping. They would (and often do) crush the competition when competing against women, and I don't think that's fair to any biological female athlete. That said, I don't think any FtM person has ever competed on the global level post-transition, so they have to choose between never being competitive against their gender, or absolutely crushing the competition against the gender they no longer identify with.
I've heard that MtF transgenders may not gain any biological advantage if they take puberty blockers and transition once they're older, and if that's true, then I'd be fine with them competing against biological women.
In general though, it seems like there can't be a good blanket rule about who should be able to compete against who that everyone agrees with. MtF people probably have too many biological differences to fairly compete against men or women, and the same goes with FtM people. That means that we either have to 1) ban transgender people from competing in sports, 2) Allow them to compete against the gender they'd like to, or 3) create a new division for trans people specifically.
Personally, I think that 1 is probably the worst option. I'd hate to see a whole group of people not allowed to compete at all, and I'd hate for an athlete to want to transition, but hold off on it because they would be banned from the sport. 2 is the second-worst option for me, just because it seems really unfair to let people compete when they are taking drugs that heavily change the body. 3 seems like the best option out of the ones I listed, but I feel like it could lead to a stigmatization of trans people as being 'others', and it would probably be really hard to regulate any kind of drug usage.