  We characterise a class of environmental noises that decrease coherent
properties of quantum channels by introducing and analysing the properties of
dephasing superchannels. These are defined as superchannels that affect only
non-classical properties of a quantum channel $\mathcal{E}$, i.e., they leave
invariant the transition probabilities induced by $\mathcal{E}$ in the
distinguished basis. We prove that such superchannels $\Xi_C$ form a particular
subclass of Schur-product supermaps that act on the Jamiolkowski state
$J(\mathcal{E})$ of a channel $\mathcal{E}$ via a Schur product, $J'=J\circ C$.
We also find physical realizations of general $\Xi_C$ through a pre- and
post-processing employing dephasing channels with memory, and show that memory
plays a non-trivial role for quantum systems of dimension $d>2$. Moreover, we
prove that coherence generating power of a general quantum channel is a
monotone under dephasing superchannels. Finally, we analyse the effect
dephasing noise can have on a quantum channel $\mathcal{E}$ by investigating
the number of distinguishable channels that $\mathcal{E}$ can be mapped to by a
family of dephasing superchannels. More precisely, we upper bound this number
in terms of hypothesis testing channel divergence between $\mathcal{E}$ and its
fully dephased version, and also relate it to the robustness of coherence of
$\mathcal{E}$.
