I will side-step the question of 'race', as you seem to be using it (like most people) in the sense that has no scientific basis, but the answer to your question goes something like this:
The amount of genetic variation in a given population is mostly the function of the geographical distance from Africa. Roughly, populations in Africa are most genetically variable as they are the oldest (they have been there for the longest time in human evolutionary history and therefore there was a long time for mutations to accumulate in their genomes). Because all the other populations in the world have originated from these original African populations, they have a subset of that original variation - they are less variable. The further you get from Africa the less variation remains (a very simplistic scenario: a group of Africans left and settled down in the Middle East; then a group of Middle Easterners went further up to Europe and further east to India; then the Indians moved further north and east etc. etc.; in each stage, only a subset of the previous variation would be travelling along).
A famous image depicting this phenomenon is in Ramachandran, S. et al. (2005). Support from the relationship of genetic and geographic distance in human populations for a serial founder effect originating in Africa. Proceedings of the National Academy of Sciences of the United States of America, 102(44), 15942–15947. doi:10.1073/pnas.0507611102 figure 1B
On the Y axis you have a measure of genetic relatedness (more relatedness = less variation) and on the X axis you have rough geographic distance that takes into account major geographic barriers (like seas etc.). Green is Eurasia and blue is Australia and Oceania. Red is variation within the groups (so, Europeans only, Australians only etc.) The variation between the populations explained by this geographic distance is close to 80% - which is huge. If I get this right, it means that most of the genetic differences between people in different populations are a simple consequence of living outside Africa. And the variation within each population is tiny (12%) compared to the between population.
This is a simplified view, because we now know that there were other species of humans that lived when our ancestors left Africa, and that thay had sex together as our ancestors spread around the globe, so a small portion of the others' DNA is still present in us today. Also this picture may be slightly different depending on what region of the genome you're looking at. But the major picture stands.
Going back to your question - "Asians" (populations with ancestry in east Asia) are on average less genetically homogenous than "Australians" or "Americans", but more than "Africans".
EDIT: A paper I like that should explain all this much better: Barbujani, G., & Colonna, V. (2010). Human genome diversity: frequently asked questions. Trends in genetics : TIG, 26(7), 285–295. doi:10.1016/j.tig.2010.04.002 link