The evolution kinetics of a bubble population driven by a uniform production of gas in a yield-stress fluid matrix is investigated in this paper. The context concerns the long time swelling of bitumen drums in which radioactive salts have been dispersed (salts suspensions in a bitumen matrix). Radioactivity generates uniform volume production of hydrogen by radiolysis of bitumen chains. Since the production rate of gas occurs on very long time scales (more than a hundred years), one needs to develop theoretical models to predict the material swelling. It has been shown previously that bitumen is a yield stress fluid and that, in the case of a motionless population (no migration due to buoyancy), the yield stress disturbs the classical scenario of nucleation, bubble growth and Ostwald ripening observed in a Newtonian fluid. In the present work, we consider bubbles nucleation and migration due to buoyancy and we show that the yield stress causes delays in the evolution of a bubble population and that it consequently modifies the macroscopic swelling of the material into the drums. (C) 2016 Elsevier B.V. All rights reserved.