<p>We study two different types of vector point processes with interacting components, introducing a migration-type effect. The first case concerns two groups which modify their states with rate functions depending on time only. This yields a representation of the vector process in terms of independent non-homogeneous Skellam processes. In the general case, the decomposition involves independent Poisson processes. The second model is a birth–death–migration vector process. In the case of the linear death–migration we show that for a fixed time instant, the vector is equal in distribution to the sum of two independent multinomial random variables. As a by-product, we derive the distribution of a pure migration process. Finally, we study the described vector processes time-changed with the inverse of Bernstein subordinators, establishing a general result concerning the relationship between fractional difference-differential equations and the probability mass function of a wider class of point processes.</p>