Fleming-Viot processes and Dawson-Watanabe processes are two classes of "superprocesses" that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the other. More specifically, we show that product moments of a Fleming-Viot process can be bounded above by the corresponding moments of the Dawson-Watanabe process with the same "underlying particle motion," and vice versa except for a multiplicative constant. As an application, we establish existence and continuity properties of local time for certain Fleming-Viot processes.