That depends on who exactly you mean with the term "the Huns". Huns, the ruling elite, were probably a distinct ethnic group (proto-Turkic and/or proto-Mongolic are two popular theories, along with others) initially ruling over people who were of their cultural kin. As they conquered their way westwards, a great variety of people were included in their "empire" (the term 'empire' is used only to point out to the vast territories they ruled over. Hun state structure was much more decentralised compared to an empire, Huns' subjects were much more free and equal compared to the subjects of an empire, and so on) from Germanic to Finn-Ugric, from Baltic to Slavic peoples, who "became" Huns after being subjected by Huns.

So, the Hun elite was a distinct ethnic group, and most probably stayed that way (names of Hun rulers, statesmen etc. suggest that, however it's always possible that people of different ethnic origins took up Hun names, like it happened with Circassian mamluks taking up Turkic names). However the Hun 'nation' was a tribal confederacy almost right from the start (when the original tribe started its conquests and ruled over/incorporated a different tribe), and its multi-ethnic nature became more and more dominant as they conquered their way westwards.

I am not able to cite any sources at the moment, might edit to add them later on. I consider myself to be well versed in the field of steppe history, so feel free to ask anything.