  We introduce a notion of compatibility for families
$(\mathcal{F}_{\ell})_{\ell}$ of bounded constructible $\ell$-adic complexes of
\'etale sheaves on schemes. For schemes of finite type over a field, this
notion is preserved by the usual six functors. We prove that the compatibility
of a family is preserved by the nearby cycles functor and by the linearized
$\varepsilon$-factors introduced recently by the author. We establish
independence of $\ell$ for the characteristic cycles and characteristic
$\varepsilon$-cycles of compatible families.
