I know that the Kullback-Leibler $D(\mu||\nu) := - \int_K\log\big(\frac{d \nu}{d \mu}\big) \, d\mu,$ over probability measures on a compact $K$ subset of $\mathbb{R}^d$, is only weakly lower semicontinuous.

What I was wondering is whether there are simple characterisations, or at least sufficient conditions for couple of measures to be points of (sequential) continuity.

In the specific problem I have in mind, I'm particularly interested in continuity with respect to $\nu$: I have a sequence $\nu_n$ weakly converging to $\nu$ and I would want to understand cases depending on $\nu$ which allow to infer that the sequence $D(\mu, \nu_n)$ converges to $D(\mu, \nu)$.