One of the basic tenets in information theory, the data processing inequality, states that output divergence does not exceed the input divergence for any channel.  For channels without input constraints, various estimates on the amount of such contraction are known, e.g., Dobrushin's coefficient for the total variation. This work investigates channels with average input cost constraint. It is found that while the contraction coefficient typically equals one, the information nevertheless dissipates. A certain non-linear function, a emph{Dobrushin curve} of the channel, is proposed to quantify the amount of dissipation. Tools for evaluating the Dobrushin curve of additive-noise channels are developed. Applications in stochastic control and uniqueness of Gibbs measures are discussed.