  In this paper, we provide tools to study the dynamics of point vortex
dynamics on $\mathbb{CP}^n$ and the flag manifold
$\mathbb{F}_{1,2}(\mathbb{C}^3)$. These are the only K\"ahler twistor spaces
arising from 4-manifolds. We give an explicit expression for Green's function
on $\mathbb{CP}^n$ which enables us to determine the Hamiltonian $H$ and the
equations of motions for the point vortex problem on $\mathbb{CP}^n$. Moreover,
we determine the momentum map $\mu:\mathbb{F}_{1,2}(\mathbb{C}^3)\to
\mathfrak{su}^*(3)$ on the flag manifold.
