The cynic in me...who is never far below the surface...wonders how many of these sponsorships come with explicit directions as to what "answers" and "truths" the studies in question are expected to "discover" in exchange for their funding.
In my experience, it would be difficult.
In fact, I don’t know that I can even think of somebody who took up one of these positions who has actually met the donor or the donor’s family.
Usually, somebody from the university office of alumni and donor relations would do the legwork in convincing the wealthy benefactor to donate their hard-earned cash. They’d come up with a name that makes the donor look good - ex.) The Richie J. Richman Fellowship in Cutting Edge Cancer Research. The money would then be deposited into the university’s accounts, with the earmark that it only be used for funding that specific position.
Then, the next academic year, whatever Department the position was going to be based in would send out an advertisement for the term position. They’d form a “hiring committee,” which normally includes members of the Department, one or two members from another university department in the same institution, and perhaps an external representative from the same field but another university. It would be extremely odd for somebody representing the donor to be included on the committee, though I can’t say for sure it has never happened.
Then, once the candidate gets the job, there wouldn’t really be a way to keep tabs on their work, aside from reading what they publish and then complaining to the university about it if the donors don’t like it. And the publications might only come out after the term fellowship is already finished.
To be honest, I suspect that most donors concern with the ongoing work ends with getting their name on the position, maybe being recognized at an alumni night or in the university magazine, or maybe getting a building or something named after them.