The partonic transverse momentum, kt , distribution plays a crucial role in driving high-energy hadron interactions. If kt is limited we have old fashioned Regge theory. If kt increases with energy the interaction may be described by perturbative QCD. We use BFKL diffusion in lnkt , supplemented by a stronger absorption of low kt partons, to estimate the growth of the mean transverse momenta ⟨kt⟩ with energy. This growth reveals itself in the distribution of secondaries produced at the collider energies. We present a simple, BFKL-based, model to demonstrate the possible size of the effect. Moreover, we propose a way to evaluate experimentally the shape of the parton transverse momenta distribution by studying the spectra of the ( D or B ) m...