Occupancy processes are a broad class of discrete time Markov chains on $\{0,1\}^{n}$ encompassing models from diverse areas. This model is compared to a collection of $n$ independent Markov chains on $\{0,1\}$, which we call the independent site model. We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.}