They can DNA trace families - not races.
So at the moment they can sort of trace human ancestry back to Africa somewhere, I think they use DNA a bit for that, and they can sort of trace migration from Africa, across India, South East Asia to Australia - and the Australian Aboriginies from - I don't know which ones - have a closer "relationship"? to the Original Africans than almost any other race. Bear in mind there's been a lot of mixing the whole way along. Until white men set up "Customs" and "Border Control", there was/is a lot of mixing between Indonesians, PNG and Northern Australians.
I think, can't remember where I read it, but Aborigines were very keen on "out crossing" to prevent "in breeding" or genetic disease ie they knew if they had babies with people very unrelated, that those babies would usually be stronger survivors than if they had babies with the next tribe over. I wish I could remember where the hell I learned that. The Arnhem Land tribes have very strict rules about where people are allowed to get their partners from - not allowed from the same tribe.
There were some stories I learned (much later than school) about various white fella explorers (might have been Charles Sturt - a very honourable fellow) being offered the local women for sex because they wanted stronger babies. And the local women / tribes were very offended when they were turned down. Of course not every traveller turned them down. I'm not sure what the current culture is. I don't think it happens as much now. Missionaries probably put a stop to it. Sort of.
What the despots can't do is make a genetic triggered disease without risking collateral damage in their own "race". They can't say for certain if a person is Chinese/Caucasion/Negro etc based on their DNA, they can only say they're related to people who live here... It's like the DNA tests on the dogs that look like Pit Bulls coming back "Significantly Chiuaua". They can't even say based on DNA - what colour a person's skin is or what their eyes look like. I think they can say the person has a "red hair gene" but not if that person actually has red hair (the gene is expressed).
Stuff what the Forensic Science TV shows say. They're not actually scientifically correct. Yet.
Nearly there but not quite. A sample of 20 people does not reliable science make.
A New DNA Test Can ID a Suspect's Race, But Police Won't Touch It
but a correlation of 176 genetic markers is pretty good.
I found another link that suggested you can mess that up with diet. But what really messes with DNA are viruses. That's how they do Genetic Modification, they use viruses. So theres Viruses (and radiation eg Sun damage) that change genes, as well as your ancestry so it's just not 100% there - yet.
With computer finger print matching - they don't go for a 100% match either, they look for a certain number of points. Though I've heard finger prints are a fairly unreliable way to get a conviction these days too.