  We study families of complex Monge-Amp\`ere equations, focusing on the case
where the cohomology classes degenerate to a non big class.
  We establish uniform a priori $L^{\infty}$-estimates for the normalized
solutions, generalizing the recent work of S. Kolodziej and G. Tian. This has
interesting consequences in the study of the K\"ahler-Ricci flow.
