Why are some research groups not so ethnically diverse? Some research groups, especially the ones at lower tier schools in the United States, contains an excessive amount of students who have the same nationality as the professor. Some Chinese professors' groups solely consists of Chinese students.

Does this go against diversity, considering the fact that Ph.D admissions are not centralized and is not done by adcom members? Is there a constraint on this implied by school to faculty?
I'm a PhD student based in Sweden, working in a group where everybody except the PhD students are German. Another group in the same building has several Russians, and a recent Russian PhD student had a Russian supervisor. So the tendency to mono-culturalism is not unique.

Probably, many institutes have statements  promoting diversity. In practice, these don't mean much, because recruitment depends on academic networks, skills of applicants, etc. There can be different reasons for a group to tend to mono-culturalism. Some reasons I can think of:


A professor heading a research group in a foreign country probably has an extensive academic network from his/her originating country. Academic networks are powerful sources of recruitment of new staff, either senior staff or PhD students (e.g. via a Master thesis supervisor recommending them). So applicants might simply be more likely to be from the same country as the head of the group, even if the head of the group is 100% honest in his/her selections.
PhD students may appreciate if they can work with a supervisor that speaks their language, particularly if they are not so comfortable speaking English. 
It's culturally easier to work in a group with several people from ones own country.


All these factors can contribute to groups in country X with group members primarily from country Y != X.

Probably there are other reasons that I didn't think of.