  We show that the higher Chow groups with modulus of Binda-Kerz-Saito for a
smooth quasi-projective scheme $X$ is a module over the Chow ring of $X$. From
this, we deduce certain pull-backs, the projective bundle formula, and the
blow-up formula for higher Chow groups with modulus.
  We prove vanishing of $0$-cycles of higher Chow groups with modulus on
various affine varieties of dimension at least two. This shows in particular
that the multivariate analogue of Bloch-Esnault--R\"ulling computations of
additive higher Chow groups of 0-cycles vanishes.
