Your DNA isn't going to change, but those ethnic groups are transient things rather than an inherent feature of reality. An ethnic group is ultimately a relative thing. Some "African-American" people might be genetically closer to the African people from whom their ancestors came than the other ethnic groups living in America, but they're not simply "African" - they are a separate diaspora. Those ethnic groups change their makeups over time.

It's not like, say, elements. 1,000 years from now, the average genetic makeup of any given ethnic group will be observably different from how it is today, but a Hydrogen atom will still be defined in exactly the same way as now. Nazis believe that races are a concrete, reified structure that doesn't shift over time - it is a "pure" and "natural" state that becomes degraded and diluted by what they term miscegenation, rather than a fuzzy construct that changes over time.

Let's say you were born in Spain and your whole family has lived there for centuries, but your family are black - say, some of your ancestors arrived from North Africa long ago. I would still call you Spanish, but a Nazi would call you African (or more likely some kind of slur). They care about the colour of your skin more than your actual background. Likewise, no matter how many generations your family has lived in a country or how well they pass, Nazis assert that if any of your ancestors was a Jew, you are also a Jew, and that as a Jew, you are inherently inferior to non-Jews and consequently a racial contaminant.