  We prove a moving lemma which implies the contravariance of Bloch-Esnault's
additive higher Chow group in smooth affine varieties and Binda-Saito's higher
Chow group (taken in the Nisnevich topology) in smooth varieties equipped with
effective Cartier divisors. The new ingredients in the moving method are
parallel translation {\em with modulus} in the affine space that involves a new
integer parameter, and Noether's normalization lemma over a Dedekind base.
