Based on this question and answer, it seems that the obligation to honor the kohein is when everyone has equal rights, the kohein gets first pick, but not everything. For instance, if a host serves a platter of steaks, the kohein gets first pick at the steak he likes, but then doesn't continue to grab the platter every time the yisrael reaches for it. He only gets the first one. Alternatively, even when there are multiple kohanim in a shul, only the first aliya goes to a kohein.
The question was raised at the end of that answer regarding if mincha and maariv are being davened back to back, does the kohein have preference for each one, or does the fact that they are together mean he only gets to go first, but a yisrael goes second?
I'd like to ask a variation: what if a kohein, a levi, and a yisrael (walk into a bar...) eat all three meals on shabbat together with no one else attending (for instance, they are roommates). Does the kohein lead the zimmun each time, or just the first? Do we view the three meals as three opportunities to bentch, and the kohein gets the first one, or do we evaluate each one individually? Most importantly, where do we draw the line between connected and disconnected events?
(This situation would often happen to me, a yisrael, if I went to someone's house for shabbat with a kohein friend, back in college. Often my kohein friend would lead zimmun Friday night and I at lunch.)