  Let $K$ be a convex subset of the state space of a finite dimensional
$C^*$-algebra. We study the properties of channels on $K$, which are defined as
affine maps from $K$ into the state space of another algebra, extending to
completely positive maps on the subspace generated by $K$. We show that each
such map is the restriction of a completely positive map on the whole algebra,
called a generalized channel. We characterize the set of generalized channels
and also the equivalence classes of generalized channels having the same value
on $K$. Moreover, if $K$ contains the tracial state, the set of generalized
channels forms again a convex subset of a multipartite state space, this leads
to a definition of a generalized supermap, which is a generalized channel with
respect to this subset. We prove a decomposition theorem for generalized
supermaps and describe the equivalence classes. The set of generalized
supermaps having the same value on equivalent generalized channels is also
characterized. Special cases include quantum combs and process POVMs.
