We obtain the asymptotic behaviour of the longest increasing/non-decreasing subsequences in a random uniform multiset permutation in which each element in {1,...,n} occurs k times, where k may depend on n. This generalizes the famous Ulam-Hammersley problem of the case k=1. The proof relies on poissonization and a connection with variants of the Hammersley-Aldous-Diaconis particle system.