  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.
