We introduce one-way LOCC protocols for quantum state merging for compound sources, which have asymptotically optimal entanglement as well as classical communication resource costs. For the arbitrarily varying quantum source (AVQS) model, we determine the one-way entanglement distillation capacity, where we utilize the robustification and elimination techniques, well-known from classical as well as quantum channel coding under assumption of arbitrarily varying noise. Investigating quantum state merging for AVQS, we demonstrate by example, that the usual robustification procedure leads to suboptimal resource costs in this case.