  We formulate a new information-theoretic principle--the shifted composition
rule--which bounds the divergence (e.g., Kullback-Leibler or R\'enyi) between
the laws of two stochastic processes via the introduction of auxiliary shifts.
In this paper, we apply this principle to prove reverse transport inequalities
for diffusions which, by duality, imply F.-Y. Wang's celebrated dimension-free
Harnack inequalities. Our approach bridges continuous-time coupling methods
from geometric analysis with the discrete-time shifted divergence technique
from differential privacy and sampling. It also naturally gives rise to (1) an
alternative continuous-time coupling method based on optimal transport, which
bypasses Girsanov transformations, (2) functional inequalities for
discrete-time processes, and (3) a family of "reverse" Harnack inequalities.
