Talking to neelk earlier today, I was informed that the things you get if you generalize multicategories to arrows with multiple codomains as well as multiple domains are called polycategories. This blog post makes an introduction to them which, being very much about logic and cut elimination, is quite near and dear to my heart. According to these slides polycategories have something to do with quantum entanglement, which is kinda weird. Here's a more detailed paper about them.
Also a couple citations about *-autonomous categories: