This paper deals with the analysis of a class of models of population dynamics with competition and kinetic interactions. Several models are proposed to describe the dynamics of large populations of individuals undergoing kinetic (stochastic) interactions which modify the states of the interacting pair. Models are characterized by time and space structure, and are motivated by recent research activity in mathematical immunology. The evolution equations are stated in terms of nonlinear integrodifferential equations which are similar to the Boltzmann equation. This paper deals with modelling and qualitative analysis of the related Cauchy problem.