Is there a situation, for example in physics or in dynamical systems, where we have a Markov chain where the transition probability between two states satisfies a law such as $$ p(y|x) = C e^{-d(x,y)} $$ where $d(x,y)$ is either a distance (as in a metric space), or a difference of some kind (for example, difference of energy density in physics)?
The Boltzmann distribution has a similar law for the relative probabilities (the Boltzmann factor), but that's not quite the same as transition probabilities.
This can be used inside of the acceptance probability for a transition in a Metropolis-Hastings run whose goal is to sample from the Boltzmann distribution. In this case you have the difference in the energies inside the exponent. Of course in M-H you need some proposal distribution as well, and the properties of the proposal distribution enter into the final acceptance probability in general.
This also comes up in systems where transition rates are modeled using the Arrhenius equation. But in this case the amount is generally not symmetric, because it is based on the difference in energy between the initial state and the transition state, which is not generally the same both ways.