Kullback-Leibler divergence lower bound

Are there any (nontrivial) lower bounds on the KL-divergence between two densities? Informally, I am trying to study problems where $f$ is some target density, and I want to show that if $g$ is chosen "poorly", then $KL(f\Vert g)$ must be large. Examples of "poor" behaviour could include different means, moments, etc.

Harder question: Any lower bounds on the KL-divergence between two mixture models (e.g. mixture of gaussians)? For example, a lower bound in terms of the component mixtures.

The only bound I have found is equation (19) in this paper, which unfortunately doesn't seem to help.

$\mbox{KL}(f||g)\geq 0$. But seriously, this is actually a really hard problem. Relevant to this topic is the area of Large Deviations Theory, specifically Rate Functions. You'll find a compendum of bounds here for example: