  The goal of this paper is to motivate a boundedness conjecture on nearby
slopes of $\ell$-adic sheaves in positive characteristic, and to prove it for
smooth curves. For a constructible $\ell$-adic sheaf, we prove the finiteness
of the set of nearby slopes associated to a given morphism. For the constant
sheaf, we prove the vanishing of nearby slopes in case of generalized
semi-stable reduction.
