In the analysis of SDE's, the famous Grönwall inequality (or other types of more advanced comparison theorems) is often useful to bound solutions by replacing e.g. the drift term by a larger, but more "transparent", drift term: e.g. such a replacement could turn a non-autonomous SDE into an autonomous one and it could help e.g. in establishing tightness of a certain family of probability measures depending on a time-parameter.
Turning to the Markov transition functions (governed by a Fokker-Planck equation), the one-sided repercussions on individual trajectories after replacing the drift-term should carry over into "one-sided" repercussions for those transition functions. Unfortunately, I don't know the techniques or the names of the techniques which mirror the use of the Grönwall inequality on the Fokker-Planck side. Any suggestions?