Asymptotic complexity of sorting using k-comparisons
As I was told by Abhishek Methuku, this has been studied already, in fact several times, see e.g. http://www.researchgate.net/publication/3042594_Sorting_n_objects_with_a_k-sorter

For $k$ large, the answer is close to $n \log_{k!}(n)$ but e.g. $k=3$ seems to be open.