  Let S be a Noetherian scheme and f:X -> S a proper morphism. By SGA 4 XIV,
for any constructible sheaf F of Z/nZ-modules on X, the sheaves of Z/nZ-modules
R^if_*F obtained by direct image (for the etale topology) are also
constructible: there is a stratification of S on whose strata these sheaves are
locally constant constructible. After previous work of N. Katz and G. Laumon,
or L. Illusie, on the special case in which S is generically of characteristic
zero or the sheaves F are constant (with invertible torsion on S), here we
study the dependency of the stratification on F. We show that a natural
"uniform" tameness and constructibility condition satisfied by constant
sheaves, which was introduced by O. Gabber, is stable under the functors
R^if_*. If f is not proper, this result still holds assuming tameness at
infinity, relatively to S. We also prove the existence of uniform bounds on
Betti numbers, in particular for the stalks of the sheaves R^if_*Z/lZ, where l
ranges through all prime numbers invertible on S.
