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Cold brown dwarfs are excellent analogs of widely separated, gas giant exoplanets, and provide insight into the potential atmospheric chemistry and physics we may encounter in objects discovered by future direct imaging surveys. We present a low resolution R $\sim$ 300 $M$-band spectroscopic sequence of seven brown dwarfs with effective temperatures between 750 K and 250 K along with Jupiter. These spectra reveal disequilibrium abundances of carbon monoxide (CO) produced by atmospheric quenching. We use the eddy diffusion coefficient (K$_{zz}$) to estimate the strength of vertical mixing in each object. The K$_{zz}$ values of cooler gaseous objects are close to their theoretical maximum and warmer objects show weaker mixing, likely due to less efficient convective mixing in primarily radiative layers. The CO-derived K$_{zz}$ values imply that disequilibrium phosphine (PH$_{3}$) should be easily observable in all of the brown dwarfs, but none as yet show any evidence for PH$_{3}$ absorption. We find that ammonia is relatively insensitive to atmospheric quenching at these effective temperatures. We are able to improve the fit to WISE 0855's $M$-band spectrum by including both CO and water clouds in the atmospheric model.