  We present a Monte Carlo simulation designed to predict the vertical velocity
dispersion of brown dwarfs in the Milky Way. We show that since these stars are
constantly cooling, the velocity dispersion has a noticeable trend with
spectral type. With realistic assumptions for the initial-mass function,
star-formation history, and the cooling models, we show that the velocity
dispersion is roughly consistent with what is observed for M dwarfs, decreases
to cooler spectral types, and increases again for the coolest types in our
study ($\sim$T9). We predict a minimum in the velocity dispersions for L/T
transition objects, however the detailed properties of the minimum
predominately depend on the star-formation history. Since this trend is due to
brown dwarf cooling, we expect the velocity dispersion as a function of
spectral type should deviate from constancy around the hydrogen-burning limit.
We convert from velocity dispersion to vertical scale height using standard
disk models, and present similar trends in disk thickness as a function of
spectral type. We suggest that future, wide-field photometric and/or
spectroscopic missions may collect sizable samples of distant ($\sim\!1$ kpc)
of dwarfs that span the hydrogen-burning limit. As such, we speculate that such
observations may provide a unique way of constraining the average spectral type
of hydrogen-burning.
