How does vertical wind change over time?
There are a number of causes, for a variety of scales.
Generally speaking, you can boil it down to convergence/divergence, as a continuity of mass problem, or buoyant motion.

You can refer to the vertical component of the Navier Stoke's equations.
$$
\frac{Dw}{Dt}=\frac{∂w}{∂t}+\vec{v}\cdot\nabla w + w\frac{∂w}{∂z} =-\frac{1}{ρ}\frac{∂P}{∂z}-g+\nu\nabla^2w
$$
Additionally, it must also be constrained by the conservation of mass.
$$
\frac{∂P}{∂t}=-\nabla\cdot(\rho\vec{v})
$$
The Quasigeostrophic Omega Equation may also provide you some insight. This equation is filtered down from the above Navier-Stokes equation, though it may not seem like it. Omega is vertical velocity in pressure coordinates.
$$
(\nabla^2+\frac{f_0^2∂^2}{\sigma∂\Phi^2})\omega=\frac{f∂}{\sigma∂p}(\vec{v}\cdot\nabla(\frac{1}{f_0}\nabla^2\Phi+f)) + \frac{1}{\sigma}\nabla^2[\vec{v_g}\cdot\nabla(-\frac{∂\Phi}{∂p})]
$$