I’m going to throw a few rambling thoughts in here. I’m not sure I’m the best qualified (where I’m from, Ireland, has had a fairly homogeneous population up to perhaps 30 years ago). There has been a lot of inward migration since then (a change, I guess, from extensive often rather involuntary outward migration), much from Europe and the EU but much from elsewhere too so, for example, my own partner is from a Chinese background, one sister-in-law is Thai (brought back from Australia by my brother where a lot of Irish people spend a year or two on working holiday visas in their youth
), though another sister-in-law is just a plain ol’ Irish girl. Anyway, thinking and having to deal with this whole race issue is fairly new here. I’m not going to get too into that.
Still, Buddhism does tend in a fairly white, educated and affluent direction in terms of membership. Though, one certainly would come across Thai and Sri Lankan etc. people far more in it than one would expect in everyday life, so it’s really hard to judge. 
I suppose Irish and white (Europeans etc.) Buddhists do tend to invariably be converts (there just hasn’t been the time for that not to be the case). That must be a significant issue. I come from a fairly typical Irish Catholic working class background; most of my relatives, parents, uncles and aunts still are. I have noticed that foreign Catholics living here do tend to fairly seamlessly slot into that whole scene (Filipinos, Polish etc. – even priests coming from African countries where missions originally went out to, now that so few Irish people become priests – a bit ironic). I guess, apart from the language, the experience is fairly similar everywhere. I can remember one or two Catholic converts from my teenage years when I was still being dragged along to mass. They were really serious about the whole thing, very quick to get involved, sign up for roles etc. I suppose when one is brought up in a religion one tends to take it more for granted, maybe be a bit more complacent about it all, not think so deeply about it. So, I guess a Catholic coming in from abroad is going to have a similar mindset and attitude in that way to Irish Catholics. I wonder if a foreign Catholic showed up in a church where everyone was a Catholic convert, would they just find it all a bit over-earnest and just a bit too much? I guess back when Jesus or Buddha were originally walking the earth, then everyone at the time was a convert. However, things probably get a bit different when a religion is there for several generations passing from parents to children. Hopefully, Buddhism in the West will get to that stage. Realistically, IMO monasticism (the role of monks and nuns) needs to become more prominent if that’s going to happen (the religion heading towards how it is practiced out in Asia): the whole individualistic retreat centre approach is only going to go so far.
I commented earlier that Buddhism does seem to tend white, affluent and educated in the West. In terms of race and whiteness (an issue raised by the OP), the reasons for that are definitely worth pondering (though I don’t feel terribly qualified to speak on that).
That does leave affluence and education too. Certainly in the US, the conditions of the “working class” (usually defined as those without a college education) has gone backwards since around the 1970s, work conditions getting more precarious, standard of living falling in relative terms (whereas the “middle class” has become more affluent relatively and the top few percentage doing really well – inequality has risen rather sharply). I don’t have much experience of US Buddhism. However, Buddhism centred on expensive retreat centres is not exactly conducive to the participation of less affluent folks. At least in Ireland, any of the more Theravada groups tend to be donation based, which probably helps. Sunyata, one of the main retreat centres in Clare in Ireland (run by a grass roots group of volunteers), doesn’t ask for anything upfront. There’s an envelope slot in the entrance hall to the main retreat hall and at some point during a multiday retreat, one can quietly slip an envelope or some money into that. If one didn’t give anything, no one will really notice. Nonetheless, things do still skew more educated and affluent on average.