If you're a member of a group that's being oppressed in a particular way you probably (if you're fairly enlightened) want two things - the eventual destruction of the system that sustains that oppression, and two - as much resistance, breathing room, and progress as you can muster in the meantime. Always (ALWAYS!) it's the people suffering alongside you in that particular manner who are keenest to the needs of the situation, most willing to take action, and best placed to help create a culture of change that's effective in the here and now, not just some theoretical future.
I've seen some fantastic cross-cultural support and activism over the years, but surely it has to be acknowledged that no lefty organisation was about to take up the fight at Stonewall, or urgently remind the world that Black Lives Matter or whatever.
I'm suspicious of the whole opposition to identity as a tool for resistance. It feels a lot to me like people feel uncomfortable replacing their existing model, in which there is a clear outside enemy oppressing the group they belong to, with a more complex model in which we all have to deal with our own small (or otherwise) part in facilitating other people's oppression.