In this paper we provide a range of examples to illustrate the general theory developed in Ref. 19, where we studied smoothing and ergodicity for infinite dimensional Markovian systems with hypocoercive type generator. We also introduce and study new models, where the framework of Ref. 19 cannot be applied as is but can be adapted to obtain improved results, by exploiting the specific structure of the generator at hand. Among such examples, we examine a system of infinitely many interacting heat baths.