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The article presents new entropic continuity bounds for conditionalexpectations and conditional covariance matrices.  These bounds are expressed in terms of the relative entropy between differentcoupling distributions.  Our approach combines Wasserstein coupling with quadratic transportation costinequalities. We illustrate the impact of these results in the context ofentropic optimal transport problems.  The entropic continuity theorem presented in the article allows to estimate  the conditional expectations and the conditional covariances of Schr\"odinger and Sinkhorn transitions in terms of the relative  entropy between the corresponding bridges. These entropic continuity boundsturns out to be a very useful tool for obtaining remarkably simple proofs ofthe exponential decays of the gradient and the Hessian of Schr\"odinger andSinkhorn bridge potentials.

Entropic continuity bounds for conditional covariances with applications to Schr\" odinger and Sinkhorn bridges