This is a sequel to the authors’ [Int. Math. Res. Not. 2015, No. 1, 1–54 (2015; Zbl 1349.14020)]. There a commutative differential graded algebra structure was produced for each integer \(m \geq 1\) on the family \(\{TCH^q (X, n; m) \}_{q, n \in \mathbb{N} }\) formed by the additive higher Chow groups of a smooth projective variety \(X\) over a perfect field \(k\) of characteristic \(\neq 2 \). By having \(m\) vary it turns out that one has a projective system of \(DGA\)’ s, this was proved to be equipped with Frobenius and Verschiebung operators satisfying all the properties required for the definition of a restricted Witt complex over \(k\), as considered by L. Hesselholt and I. Madsen [in: Homotopy methods in algebraic topology. Proc. AMS-IMS-SIAM joint summer research conference, Boulder, CO, USA, 1999. Contemp. Math. 271, 127–140 (2001; Zbl 0992.19002)], see also K. Rülling [J. Algebr. Geom. 16, No. 1, 109–169 (2007; Zbl 1122.14006)]. In [loc. cit.] a scholium was stated, to the effect that if the moving lemma could be proved to hold for the additive higher groups in case of smooth affine or quasi projective varieties then the main results there would extend also in this setting. Such a moving lemma was announced by W. Kai [“A moving lemma for algebraic cycles with modulus and contravariance”, Preprint, arXiv:1507.07619], whose proof is illustrated for completeness also in the present paper. In this way the authors can continue their program, according to their expectations they show that for a smooth \(k\) scheme \(X = \mathrm{Spec} (R)\) which is essentially of finite type the family \(\{TCH^q (X, n; m) \}_{q, n, m }\,\) has the structure of a restricted Witt complex, both over \(k\) and over \(R\) . Moreover they investigate also certain multivariate additive higher Chow groups \(\{ CH^q(X[r]|D_{\underline {m}},n)\}_{q, n \geq 0}\) with modulus \(\underline{m}= (m_1, \cdots, m_r)\), which are a particular but interesting case of the groups studied by F. Binda and S. Saito [“Relative cycles with moduli and regulator maps”, Preprint, arXiv:1412.0385]. It is proved that the multivariate groups form a differential graded module over \(\{TCH^q (X, n; m-1) \}_{q, n \in \mathbb{N} }\,\), with \(m = |{\underline {m}}|\).