Its complicated. Within a university there is some revenue sharing for the "have-nots" as those bring in the big research grants are paying 55%+ overhead back to the university, some of this is passed back down to departs where "have-nots" may be able to access funds. However most is probably chewed up in admin expenses, salary, campus O&M etc. If the grant is certain Fed govt funding the overheard rate is not as large at say Land Grants cause the Feds established/funded Land Grants over the long run.
There is often collaboration among universities some within conferences say B10 where 3 to 5 universities (faculty within related depts) work together on joint research and thus share the overhead or admin burden a sorta "have-not" member could kindve be a free rider.
Another example a recent very large NSF project where Illinois was the lead and included Brown, Stanford and Arkansas. Seems like a odd mix, NSF sometimes wants a little diversity and yeah there's a bit of wokenes in Federal funding.
Another model, preferred by some, is go directly to industry for research on industry needs etc, gonna pay the full overhead rate, grants and contract will waste a lot of time worrying about every little IP issue but more freedom to partner with others on campus or at other institutions, course industry wants results not just another publication.
Ok that was my short one cup of coffee response.