  The celebrated Erd\H{o}s-Ko-Rado Theorem and other intersection theorems on
families of sets have previously been generalised with respect to families of
permutations. We prove that many of these generalised intersection theorems for
permutation families and related families naturally extend to families of
matchings of complete $k$-partite $k$-graphs. Our results may be seen as
surprising and strong generalisations of the Erd\H{o}s-Ko-Rado Theorem, and our
proofs rely merely on relatively simple inductive arguments.
