Empires are diverse by their very nature. The need of conquest leads them to meet new ethnic groups and incorporate them within their empires. There is a lot around if you look. Russia, whether you consider it to be a colonial empire today or not, is still diverse. During the USSR the Russians at most consisted of 40% of the population. China officially have 56 minority groups though unofficially the number is higher. The British Empire was outnumbered by its subjects solely in British India.

In more historical terms the Mongol Empire held a multitude of groups which outnumbered the Mongols themselves. Persians, Turkic groups, Chinese, Uyghur and other were commonly referred to in the literature.

I do not know what sort of evidence you're looking for. If you want ethnic DNA % in a population as some in that debate want, then I wouldn't know where or even how to look nor could anyone I guess except for the accounts that exist. No empire, with the exception of those on mainland Europe (discounting Spain, Italy and Sicily) solely was composed of one ethnicity. As soon as any European empire became a colonial power there existed a number of different ethnic groups within their boundaries.

I really don't know how to give you sources for this topic as it is incredibly large question.