We consider a family of stadium-like billiards with time-dependent boundaries. Two different cases of time dependence are studied: (i) the fixed boundary approximation and (ii) the exact model which takes into account the motion of the boundary. It is shown that when the billiards possess strong chaotic properties, the sequence of their boundary perturbations is the Fermi acceleration phenomenon which is three times larger than in the case of the fixed boundary approximation. However, weak mixing in such billiards leads to particle separation. Depending on the initial velocity three different things occur: (i) the particle ensemble may accelerate; (ii) the average velocity may stay constant or (iii) it may even decrease. (C) 2010 Elsevier B.V. All rights reserved.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)