  In Bayesian statistics, a continuity property of the posterior distribution
with respect to the observable variable is crucial as it expresses
well-posedness, i.e., stability with respect to errors in the measurement of
data. Essentially, this requires to analyze the continuity of a probability
kernel or, equivalently, of a conditional probability distribution with respect
to the conditioning variable.
  Here, we give general conditions for the Lipschitz continuity of probability
kernels with respect to metric structures arising within the optimal transport
framework, such as the Wasserstein metric. For dominated probability kernels
over finite-dimensional spaces, we show Lipschitz continuity results with a
Lipschitz constant enjoying explicit bounds in terms of Fisher-information
functionals and weighted Poincar\'e constants. We also provide results for
kernels with moving support, for infinite-dimensional spaces and for non
dominated kernels. We show applications to several problems in Bayesian
statistics, such as approximation of posterior distributions by mixtures and
posterior consistency.
