<p>It’s faculty that apply for research grants, not the university. In fact, faculty usually can take their grants with them when they leave and there are also special faculty development grants for faculty at undergrad institutions,. However, you have to ask yourself whether you want to have research grants for, say, cancer research, go to the faculty who are most likely to produce results, regardless of whether they work at a public or private institution. And, no, most grant applications are not funded, even if they are from Princeton.</p>