  For a planar domain $\Omega$, we consider the Dirichlet spaces with respect
to a base point $\zeta\in\Omega$ and the corresponding kernel functions. It is
not known how these kernel functions behave as we vary the base point. In this
note, we prove that these kernel functions vary smoothly. As an application of
the smoothness result, we prove a Ramadanov-type theorem for these kernel
functions on $\Omega\times\Omega$. This extends the previously known
convergence results of these kernel functions. In fact, we have made these
observations in a more general setting, that is, for weighted kernel functions
and their higher-order counterparts.
