Is there such a thing as a first right-of-refusal for an equity stake or C-level position?

Is there a type of agreement which articulates a consultant's first "right of refusal" for an equity stake and C-level position if/once a company gets Series B funding? Perhaps based on consultant performance or other metrics? Is there precedent for this? Is there a name for such an agreement? Would it be binding? I am in discussions with a 5-year old consulting company which is launching an arm of their business in which I have expertise and that also overlaps with a concept which I've been exploring over the past few months. In this case, I would lead the effort to launch this arm of their business under their brand (part-time). This would include business development, bringing on the consulting team and be on a part-time basis. They just raised Series A; they plan to seek a Series B round next year. If they get Series B, they say that I could be brought in f/t with an ownership stake and proper pay, assuming that all goes well with my consulting stint. I just don't know if there's a type of agreement which articulates first right of refusal for this equity stake and/or C-level position if they get Series B funding. If it exists, is it binding and what is it called?