environments, or to highlight the influence of few key investors in the stock market (see the discussion in [@bongini2016optimal Introduction]). In a multi-population setting, source (or sink) terms can be added to the model, to account for the ”birth” and ”death” rate of a single population (see for instance [@Rossi-review Sections 4-5]). Furthermore, even in the case where the number of agents remains constant along the evolution, the possibility of having some individuals switching from one population to another has been considered [@thai].

As a matter of fact, exchange rates among populations are a common feature in several applications. For instance, in models of chemical reaction networks a particle may change its type as a result of the interaction with the others, at a stochastic rate which may also depend on its position [@LLN2019; @CRN; @O1989]. Another relevant example comes from social dynamics, where transition rates appear in some kinetic models of opinion formation in the presence of strong leaders, as the one proposed in [@BMPW2009], inspired by the earlier contribution [@Toscani2006]. In this model, the opinion of the agents, described by their position in the state space, evolves because of the exchange of opinion with the others. This is 