Suppose a faculty member has external funding for PhD students (who would otherwise have TA positions). How do they decide which of their PhD students get funding? For instance, I've seen faculty who give priority to their most senior students or who rotate funding between all students. Are these choices at the discretion of the PI, or does the department/university/funding agency have some fine-grained control?
In particular, are there constraints with regard to non-discrimination in the same way there are when hiring students in the first place?
I'm in the US, but would be interested in hearing responses from different academic cultures as well.