Of course, anyone who concerns him or herself with equality has to think about King. Thinking about him invariably brings to mind the man who is often thought to be his opposite: Malcolm X, who was also gunned down.
Most people who are working, or have worked, for the betterment of their group of people see King as an icon. The fact that he faced the institutionalized (and, too often, legal) discrimination of his time is reason enough to venerate him. However, I think that it's often assumed that King would be in support of some cause or another. The same people might have wanted to claim Malcolm X as an ally, except that they see him as someone who pursued his ends through violence, and advised others to do the same.
I think it might be fair to assume that King would favor LGBT equality. However, I'm not so sure of what he'd think of same-sex marriage, or even legal protections for transgender people. After all, he was ordained as a minister in a conservative Protestant denomination. Then again, his views on LGBT equality might have evolved had he lived longer. There was evidence that some of his views, or at least his focus, was starting to change in the days before his assassination. He had treated the fight for equality mainly as a legal and moral issue. However, it is said that by the end of his life, he was starting to think about the economics of the issue more. Perhaps he understood, as James Baldwin said more than half a century ago, that no community can hope to improve its lot when even the local supermarket isn't owned by someone in the community and the profits are going to some faraway place.
Malcolm X understood that from the beginning of his days as an activist. He said that, in essence, African-Americans had to develop their own economy, which would be separate from the larger American/Western economy. He did not want, or expect any assistance from the government or any other established institution. Ultimately, I believe that he had the right idea: Groups of people, if they want to better themselves, ultimately must establish their own stores, banks, schools and whatever. However, that wasn't possible in his time because African-Americans and other marginalized communities didn't have the resources necessary to bring about such a system.
King was beginning to understand that. And, after the journey to Mecca he took the year before he was killed, Malcolm was starting to realize that in order to bring about economic, not to mention ethical and spiritual, justice, unity rather than separatism was needed. So he started to see the need for a universal brotherhood that would include alliances and friendships with sympathetic whites and people of other races. That, of course, was Martin Luther King's territiory.
Would they have joined forces? (Interestingly, their widows became very close friends.) Would their alliances and brotherhoods have include LGBT people. Somehow the notion that they would have done those things seems not so far-fetched.