Let $P,Q,R$ be probability measures on the real line.
If these are discrete, we can show
\begin{equation}
D\_{\mathrm{KL}}(P\ast R\,\Vert\,Q\ast R)\le D\_{\mathrm{KL}}(P\,\Vert\,Q)
\end{equation}
by using the log sum inequality, where $D\_{\mathrm{KL}}(P\,\Vert\,Q)=\int \log\left(\frac{dP}{dQ}\right) dP$ is KL divergence and $\ast$ is convolution.
Is the above inequality true when $P,Q,R$ are not necessarily discrete?



KL divergence and convolution of distributions
The KL divergence cannot increase after passing both distributions through the same Markov kernel (in your case, convolution with $R$). This is an immediate consequence of the data processing inequality:
<https://en.wikipedia.org/wiki/Data_processing_inequality>


