Even though we talk about race and diversity, what's often at stake is actually representation. If I'm writing for a general US audience, that means we're looking at 12% blacks, 17% Hispanic/Latinos, and smaller numbers of Asians and Muslims. So two questions:
1) Beginning with strict historical realism, how common would these groups be in different parts of Europe?
2) And what are the most believable and easy-on-the-story ways to relax strict realism and increase the prominence of these groups in a European setting?
Worthwhile questions. I pretty much dodge them in Altearth, partly because I know just enough of the literature about blacks in medieval Europe to know I either need to read much more or I need to sit down, fold my hands, and be quiet.
As for Hispanic, that's easier, I think, because they didn't exist in medieval Europe, not in any way that is meaningful in the modern world. There were Castilians and Catalans and Galicians, but to me the whole vocabulary around Latino and Hispanic implies New World. I'd be much more concerned with not mixing up the Spanish stuff--treating Navarre and Castile and Aragon as all "Spanish."
There are almost no Asians, unless you want to include the Middle East as "Asian." But you said Muslim explicitly, and that one's more complicated. There are the Moors, who occupied a fluid area in Iberia from the 700s right up to 1500, but there was also a large Muslim population in Sicily for many centuries, and isolated communities elsewhere. Including one in the Alps! I can't give any references; my very limited knowledge is all occasional and tangential.
Then you ask the big question. How can a writer increase the prominence of a particular group? Having recently read Children of Earth and Sky, you could take that path--simply set your story somewhere in the Balkans, or in Venice or Genoa or Pisa, and you have a ready-made stew. Go bigger and set your story somewhere in the Middle East. Doesn't have to be Cairo or Jerusalem; it could be Cyprus or Georgia or Armenia. The Empire of Trebizond. You're in for some major reading, but those are settings seldom tapped.
Another approach would be to come up with a reason why some group is in a place where they weren't historically. Real historical parallels could be the Jews coming to Venice from Spain, or the settlement of Muslims in the Italian town of Lucera, done by Emperor Frederick. So it could be done. All you need do is come up with a world or story reason why your particular group were in that particular place. Just for reference, in the late MA, the wealthy in Venice kept personal slaves. Being a nation of lawyers, the Venetians argued that the prohibition against slavery applied only to enslaving Christians, so they bought slaves in the Arab slave markets--mostly Slavs, but some African blacks.
Pursuing that last, slaves taken in war could provide an excuse. Some great victory that resulting in disappointing plunder, so they took whole families of slaves. Then you could set a story about how those people rebelled, or were ground down, or eventually won freedom and everyone lived in peace and understanding ... whatever story you wanted to tell.
Overall, though, I tend to avoid this sort of thing. Matters of race (or other modern hot-button issues) are too important, imo. The stories about such things are both more powerful and more influential if they are set in the real world. I'm currently reading The Kite Runner. I don't think that story would have benefited from being set in a fantasy world.