  For a mixture of interacting Bose gases initially prepared in a regime of
condensation (uncorrelation), it is proved that in the course of the the time
evolution observables of disjoint sets of particles of each species have
correlation functions that remain asymptotically small in the total number of
particles and display a controlled growth in time. This is obtained by means of
ad hoc estimates of Lieb-Robinson type on the propagation of the interaction,
established here for the multi-component Bose mixture.
